Subset Topological Spaces and Kakutani s Theorem

Transcription

1

2 MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1

3 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered from: EuropaNova ASBL Clos du Parnasse, E 1, Bruxelles Belgum E-mal: nfo@europanova.be URL: Peer revewers: Dr. Stefan Vladutescu, Unversty of Craova, Romana. Professor Paul P. Wang, Ph D, Department of Electrcal & Computer Engneerng, Pratt School of Engneerng, Duke Unversty, Durham, NC 778, USA Sad Broum, Unversty of Hassan Mohammeda, Hay El Baraka Ben M'sk, Casablanca B. P. 7951, Morocco Many books can be downloaded from the followng Dgtal Lbrary of Scence: SBN-1: EAN: Prnted n the Unted States of Amerca

4 CONTENTS Preface Chapter One BASC CONCEPTS 7 Chapter Two MOD SUBSET SEMGROUPS UNDER AND 9 Chapter Three MOD SUBSET SPECAL TYPE OF NTERVAL TOPOLOGCAL SPACES 19

5 Chapter Four MOD SUBSET TOPOLOGCAL SPACES ON THE MOD PLANE R N (M) AND KAKUTAN S THEOREM 191 FURTHER READNG 5 NDEX 9 ABOUT THE AUTHORS 7

6 PREFACE n ths book authors for the frst tme develop the noton of MOD natural neutrosophc subset specal type of topologcal spaces usng MOD natural neutrosophc dual numbers or MOD natural neutrosophc fnte complex number or MOD natural neutrosophc-neutrosophc numbers and so on to buld ther respectve MOD semgroups. Later they extend ths concept to MOD nterval subset semgroups and MOD nterval neutrosophc subset semgroups. Usng these MOD nterval semgroups and MOD nterval natural neutrosophc subset semgroups specal type of subset topologcal spaces are bult. Further usng these MOD subsets we buld MOD nterval subset matrx semgroups and MOD nterval subset matrx specal type of matrx topologcal spaces. 5

7 Lkewse usng MOD nterval natural neutrosophc subsets matrces semgroups we can buld MOD nterval natural neutrosophc matrx subset specal type of topologcal spaces. We also do buld MOD subset coeffcent polynomal specal type of topologcal spaces. The fnal chapter manly proposes several open conjectures about the valdty of the Kakutan s fxed pont theorem for all MOD specal type of subset topologcal spaces. We wsh to acknowledge Dr. K Kandasamy for hs sustaned support and encouragement n the wrtng of ths book. W.B.VASANTHA KANDASAMY LANTHENRAL K FLORENTN SMARANDACHE

8 Chapter One BASC CONCEPTS MOD SUBSET SPECAL TYPE OF NTERVAL TOPOLOGCAL SPACES n ths chapter we just menton how the MOD subset specal type of topologcal spaces, MOD natural neutrosophc specal type of topologcal spaces, MOD nterval subset specal type of topologcal spaces and MOD nterval natural neutrosophc subset specal type of topologcal spaces are bult usng Z n, Z n, Z n g, Z n h, Z n k and C(Z n ), Z n g, Z, C (Z n ), Z n k, Z n and Z n h, [, n) C([, n n)), [, n)g, [, n) h, [, n) and [, n) k and [, n), [, n) g, [, n) h, [, n) k, C ([, n)) and [, n) respectvely are used n buldng these MOD specal type of topologcal spaces. We can bult MOD topologcal spaces usng and whch can be of fnte or nfnte order. The valdty of the Kakutan theorem remans a open problem n these spaces also. On these MOD nterval subset specal type of topologcal spaces usng P([, n)) or P( [, n)) or P([, n) g) or P([, n) g )or P([, n) )or P([, n) )or P([, n) k ) or P([, n) h) or P([, n) h ) a study of Kakutan s fxed pont theorem s suggested.

9 8 MOD Natural Neutrosophc Subset Topologcal For n many cases one s forced to accept the exstence of fxed ponts however n some cases the other propertes of Kakutan s theorem many not be true n general. n fact the study or verfcaton of Kakutan s theorem n case of these MOD subset specal type of topologcal spaces constructed n ths book happens to be a challengng problem for these spaces not only use or whch s used n the classcal topologcal spaces but these MOD subset specal type of topologcal spaces makes use of the operatons + and, + and, + and, and and apart from and. So several structural changes occur for all operatons but do not n general yeld dempotents. What we are satsfed wth s the closure axom and the basc defnton for the collecton of subsets to be topologcal spaces, whch s why we call these topologcal spaces as MOD subset specal type of topologcal spaces. Fnally we do not always use subsets from reals or complex or ratonal numbers they can be matrx subsets or matrx wth subset entres or t can be polynomal subsets or polynomals wth subset coeffcents/ entres from the above sad test. However under the two operatons taken from the set of operaton {+,,, } we see they are MOD subset specal type of topologcal spaces. f Kakutan s theorem or a modfed form of Kakutan s theorem s proved for the MOD subset specal type of topologcal spaces for certan they wll hold good n case of MOD matrces subset specal type of topologcal spaces and MOD polynomal subset specal type of topologcal spaces for when we say subsets t can be anythng need not always take values from R or C or Q and further the operatons need not be or t can be + or also or a combnaton for any two from the four operatons. So such study s nterestng and nnovatve, throw many challenges to researchers n topology.

10 Chapter Two MOD SUBSET SEMGROUPS UNDER AND n ths chapter for the frst tme we ntroduce the operatons of the unon or the ntersecton of the MOD subsets of S(Z n ) (or S(C(Z n )) or S( Z ) or S(C (Z n )) or S(Z n g) or S(Z n g ) or S(Z n h) or S(Z n h ) or S(Z n k) or S(Z n k ) or S(Z n )or S(Z n )). We wll frst llustrate ths stuaton by examples. However f s to be a closed operaton on S(Z n ), we have to nduct the element, for f A, B S(Z n ) then f A B = then to have {S(Z n, } to be a semgroup so we have to nduct wth S(Z n ) then only {S(Z n ), } s a semgroup. We wll llustrate ths by examples. Example.1. G = {S(Z ), } s defned as the MOD semgroup. Let A = {,, } and B = {1, } G, A B = {,, } {1, } = {, 1,, } = Z. Let X = {} and Y = {1, } G, X Y = {} {1, } = {, 1, }. n

11 1 MOD Natural Neutrosophc Subset Topologcal Thus G s a semgroup or a semlattce. {,1,,} {,1,} {1,,} {,,} {,1,} {1,} {,} {,1} {} {1} {} {,} {} {1,} {,} Fgure.1 So G s a semlattce gven n fgure.1. Example.. Let H = {S(Z ), } be the semgroup. o(h) = 1. H = {{}, {1}, {}, {, 1}, {, },{1, }, {1,, }}. H s nfact a semlattce. Example.. Let W = {S(Z ), } be the MOD semgroup (or semlattce) of order 1. Ths has subsemgroups for take A = {}, then A A = {}. Thus every sngleton set s a subsemgroup. Let B = {, 5} W; B B = {, 5} = B. Hence every element of W s a subsemgroup of order one. Let A = {,, 5} and B = {, 9, 8, 1, } W. A B = {,, 5} {, 9, 8, 1, } = {,, 5,, 8, 9, 1, }. Hence P = {A, B, A B} s a subsemgroup of order three.

12 MOD Subset Semgroups Under 11 Thus every par of subsets A, B n S(Z ) wth A B or B A generates a subsemgroup of order three gven by {A, B, A B}; ths s true for every A, B S(Z ). Let A = {, 5, 7, 8, 9, 1, } and B = {, 5, 7, 8, 9, 1,,, 1, 5, 8, 9} S(Z ). A B; so A B = B. Thus M = {A, B} W s a subsemgroup of order two. Hence one can get subsemgroups (subsemlattces under ) of any desred order bounded by 1. However f s not ncluded n S(Z ) we wll not be n a poston to defne the operaton. Thus S(Z ) = P(Z ). We can defne and under, P(Z n ); n < s a semgroup or a semlattce. We wll gve one or two examples of them before we make a comparson between (P(Z n ), ), {P(Z n ), }, {P(Z n ), +} and {P(Z n ), } as semgroups. Example.. Let S = {P(Z 1 ), } be the MOD semgroup whch s also a semlattce. We see every element x n S s a subsemgroup of cardnalty one. For f x S; x x = x hence the clam. Let A = {,,, 5} and B = {,, 8} S. A B =, however {A, B} s not a subsemgroup for {A, B}. But P = {, A, B} S s a subsemgroup of S of order. Let A = and B = {9, 8,, } S; clearly A B =, so {A, B} s a MOD subsemgroup of order two.

13 1 MOD Natural Neutrosophc Subset Topologcal Let P = {,,,, 8} and Q = {, 5,, 9, } S. P Q = {,, } S. Only {P, Q, P Q} s a MOD subsemgroup of S. Let A = {, 5,, }, B = {,, 9, } and C = {, 7,,, 1} S. We see M = {A, B, C} s not a subsemgroup of S. We now fnd A B = {, }, A C = {} and B C = {, }. Thus M s not a subsemgroup however (A C) {B C} = {} = (B C) (A B) and {A B} {A C} = {}. Thus we can complete a subset of S whch s not a subsemgroup nto a subsemgroup [ ]. Here we denote by M C = {A, B, C, A B, A C, B C, (A B) (A C), (A B) (B C), (A C) (B C), (A C) (B C)} = {{}, {,}, {,}, {,, 5, }, {,,, 9}, {, 7,,, 1}}. We call M C the MOD subsemgroup completon of the subset M [, ]. We see t s always possble to complete any subset nto a subsemgroup after a fnte number of steps. n vew of all these we have the followng result. Theorem.1. Let P(Z n ) = {S } be the power set of Z n. Let S = {P(Z n ), } be the MOD semgroup. ) o(s) = n. ) S has MOD subsemgroups of all orders t; 1 t < n. ) All subsets of S can be completed to get a subsemgroup n a fnte number of steps. Proof s drect and hence left as exercse to the reader.

14 MOD Subset Semgroups Under 1 Next we show how to descrbe + and on P(Z n ), the power set of Z n by some examples. Example.5. Let {P(Z 9 ), +} = S be the MOD semgroup. We defne + A = A. Clearly A for any A. So Z 9 s dfferent from for can be n some of the subsets of Z 9. Hence by no means we can call the addtve dentty. s only a notaton to show a set s empty. When s added to A, that s nothng s added to A, nothng happens to A remans as A. Also + = wth ths {S(Z 9 ), +} s a semgroup of order 9. f A = {,,, 5} and B = {1,,, 5} S, ten A B = {, 1,,, 5,, } A B = {5} A + B = {,, 1,, 7, 8,, 5} We see all the three equatons, and are dfferent. Hence all the MOD semgroups (S, +), (S, ) and (S, ) are dfferent semgroups wth same cardnalty. nfact {S, } = {P(Z 9 ), } and {S, } = {P(Z 9 ), } are both dempotent semgroups. Now we proceed onto defne on P(Z n ) the product operaton we defne A = A for when A s product wth nothng. A remans unaffected. t s mportant to keep on record that s the nothng of the real number system lkewse the empty set s the nothng of subsets of any unversal set U. So t s left for one to defne = f we work n the subsets of numbers and {} + = {}.

15 1 MOD Natural Neutrosophc Subset Topologcal n ths book we use only ths conventon. For bascally MOD number are from Z n, Z or [,n) or [, n); n <. n Keepng ths n mnd we gve examples of MOD subset semgroups under product on P(Z n ). Example.. Let S = {P(Z 15 ), } be the MOD subset semgroup under product. f A = {, 5, 1,, 7} and B = {, 5, 9,, 11, 7, 1} S to fnd A B and compare t wth A + B, A B and A B. Now A B = {, 5, 1,, 7} {1,,, 5, 7, 9, 11} = {, 5, 1,, 7,, 1, 1, 1,, } A B = {, 5, 1,, 7} {1,,, 5, 7, 9, 11} = {5, 7} A B = {, 5, 1,, 7} {1,,, 5, 7, 9, 11} = {, 5, 1,, 7, 1,,, 9, 11} A + B = {, 5, 1,, 7} + {1,,, 5, 7, 9, 11} = {1,,, 5, 7, 9, 11,, 1,1,, 8, 1, } V t s clear the four equatons,, and V are dstnct and thus the four MOD semgroups are dfferent n ther algebrac structures. Further t s mportant to keep on record that for a topologcal space n general we need only two closed bnary operatons on the collecton of subsets. Also the subsets collecton n our case s fnte. Thus we can have dstnct MOD topologcal spaces usng the power set; P(Z n ); n <.

16 MOD Subset Semgroups Under 15 We can have {P(Z n ),, } = T o, {P(Z n ),,+} = T, {P(Z n ),,} = T, {P(Z n ),, +}= T, (P(Z n ),, } = T and T = {P(Z n), +, whch we choose to call as specal type of MOD subset topologcal spaces or MOD subset topologcal spaces of specal type. All the sx topologcal spaces are dstnct further o(p(z n )) = n only a fnte collecton of subsets. We have defned specal type of subset topologcal spaces [1]. When we say they are subsemrng of type. t s mportant to note that as we do not need any relaton between the operators on a topologcal spaces as we do not fnd relaton connectng + and or and or + and or and and so on. n case of type subset semrngs subsets are taken from semrngs and n case of type semrngs subsets are from a rng. t s mportant to note that these MOD specal type of topologcal spaces are of fnte cardnalty. Such types of specal type of subset topologcal spaces are dscussed elaborately n [1]. We have also gven subspaces usng matrx subsets. Hence we proceed to descrbe MOD natural neutrosophc specal type of subset topologcal spaces usng

17 1 MOD Natural Neutrosophc Subset Topologcal subsets of h. Z n, C (Z n ), Z n Z n g Z n k and Z n On these subsets we can defne four operatons + or or or and under all these operatons these subsets are only semgroups. We usng these subset semgroups buld S. MOD natural neutrosophc specal type of subset topologcal spaces s defned n an analogous way as S(C(Z n )) or S(Z n g) or S(Z n h) or S(Z n k) or S(Z n ). We wll llustrate ths stuaton by examples. Example.7. Let S = {collecton of all subsets from the MOD natural neutrosophc semgroup of Z, } n whch usual zero 1 s domnated that s a = for all a T o = {S = S{},, }. Z 1. {S,, } = T, T = {S,, } and T o = {S,, } are the three MOD natural neutrosophc subset specal type of topologcal spaces assocated wth the semgroup { Z 1, } , +, 1 + 1,, 8, 1 + } and B = {1, + 9, 8,, } T o. Let A = {,, A 1 5, 1 +, 1 +1,, 8, 1 + } {1, 1, , ,, } = {, 5, +, +1,, 8, +, 1,, 8 +9, } A B = {, 1 5, 1 +, 1 +1,, 8, 1 +} {1, 1, , 8,, } = {8, }

18 MOD Subset Semgroups Under 17 Take the same A, B T. A B = {, 1 5, 1 +, + 9, 8,, } = {, 1, 1 1 8, ,, 8, + 1 } {1, 1, 1 +, +1,, 8, + 1, 8 +, + 1, 1 + 1, 1,,, , ,, 1 +, , 8 + 1, 8 + 1, + 1, 1 +, + 1, Thus all the three equatons are dstnct so these MOD natural neutrosophc specal type of subset topologcal spaces are also dstnct. We see the only common property between these MOD topologcal subset spaces s that they have same cardnalty apart from that all the three are dstnct. T o beng the ordnary MOD natural neutrosophc topologcal space. t s mportant to note that T o s unaffected by any we use, n < the only factor beng that f n s a prme the n only natural neutrosophc element s, but n case when n s not a prme we have several elements whch are MOD natural neutrosophc elements and they are assocated wth T o, T and Z n T further the product operaton also contrbutes to many other propertes. We gve an example n whch many MOD natural neutrosophc elements are present. Example.8. Let S = {S( Z 1 ), } be the MOD natural neutrosophc subset semgroup under the MOD natural neutrosophc zero domnated product. Let S = S {}; T o = {S,, T = {S,, } and T = {S,, } be the three MOD natural neutrosophc specal

19 18 MOD Natural Neutrosophc Subset Topologcal type of subset topologcal spaces but usng the MOD natural neutrosophc under product subset semgroup, whch s MOD natural neutrosophc zero domnated. B = { 1 1, 1} S. Let A = { , + +, + 1, + 1 A B = { {,, 1,, 1, + 1 9, 1 } = {, 1,,,, 1, , } A B = { + 1 1, +, 1 1 +, + + 1, ,, 1, } and, 1 +,,,, 8 1, +, ,, 1, }, , + 1, + 1, 1 + 1, + 1, + 1, + 1,, 1, } {,, 1,, 8 1, + 1 9, , + 1, + 1, } = {, 1, + 1 } 1 + +, + 1 A B = { + 1, , , + 1,,, 1, + 1 } {1,,,,, + 1, 1 + 1, } = { + 1, + 1,, 1, + 1, , 8 + 8, 8 + 1, , + 1, , 8, +, 1,,, 1, 1 + 1, 1, , , 1, 8 + 1, + + 1, , , 1, 8 + 1, , , 1 + 1, + 1, 1, + 1, , , 1 + 1, , , , } Clearly, and are dstnct hence the three MOD natural neutrosophc specal type of subset topologcal spaces related to the MOD natural neutrosophc subset semgroup under MOD natural neutrosophc zero domnated product are dstnct

20 MOD Subset Semgroups Under 19 Now we fnd A B for the above sad A and B n whch the product s the usual zero domnated product A B = { , + 1, + 1,, 1, } { 1 + +, + 1, + 1, 1 + 1, + 1,,,, 1,1} = { , , , ,, , 1 + 1, , + 1, , , , 1, 8 + 1, , , 1 + 1, , , , 1, 8 + 1, , 8 + 1, , 8, , 1, + 1,, 1, , + 1,, , 8 + 1, } t s easly verfed and V are dstnct. Hence the product n whch s domnated gves a dfferent value from the product n whch MOD natural neutrosophc zero s domnated. Next we proceed onto express the MOD natural neutrosophc specal type of a subset topologcal spaces usng the MOD subset semgroup under + gven by {S(Z n ), +}; n < by some examples. Example.9. Let S = {elements from the MOD natural neutrosophc subset semgroup S( Z ), +}. Let S = {S {}}. T o = {S,, }, T = {S, and T = { S, be the MOD natural neutrosophc specal type of subset topologcal spaces assocated wth the MOD natural neutrosophc subset semgroup {S(Z 18 ), +}. Let A = { V, ,,, 5, B = {, 1,, 18, 1, 8, , }T } and

21 MOD Natural Neutrosophc Subset Topologcal A + B = { , ,,, 5, } + {, 1,, 1, 8, 18, , } = { , ,,, 5, , , ,, 7, , , , 1, 1, 15, , , , 8, 1, , , 18, +, , , , , , , , , , , , , , , } 9 1 A B = { , ,,, 5, } {, 1,, 1, 8,, 18 +5} = {, } A B = {5 + 9, + +,,, 5, + 1 } 18, } = {, 1,, 1, 8, 5, , , 18, } We see all the three equatons, and are dstnct so the three spaces T o, T and T are also dstnct. Example.1. Let S = {Elements from the MOD natural neutrosophc subset semgroup {S( Z ), +}} be the MOD natural neutrosophc specal type of subset topologcal spaces usng the MOD natural neutrosophc subset semgroup under +. S = {S = {S,, }. = { + 7, Let A = { ,,, 5, 8, + 1, 1, 1, 1, 5,,, ,, 1, } and B + } T o. A,,, 5, 8,, 1,, 19 + } + { + 7, + 1, 1, 1, 1, 5,,,, 19 +, 7 +, + } = {8 +, +1, 1, 1, 5,,,,, 5, 8,, 1, + }

22 MOD Subset Semgroups Under 1 A B = {8 +,,, 5, 8,, 1,, 19 + } { + 7, 1 +, 1, 1, 1, 5,,, + } = {1} Let A, B T (or T ). A + B = {8 +, +, 1, 1, 1, 5,,, + = { ,, +,, 5, 8,, 1,, , 1 +, + }, +, 18 +, , 7,, 11 +, 9 +, 9 + } + {7 +, 9, +, 1 +,, 1 +, +, 1 +, 1,,, 7 +, +, +, 1, 15, 18, 1, 11, +, 1, 9, 9 +, 5,, 5 +, +, 1 +, 1, 17,, 1, 1, 1, 1,, 11, 1 +, }, 1 +, 5,, 9, 5 +, 1,,, +, 1 +, Clearly, and are dstnct, hence the three MOD natural neutrosophc specal type of subset topologcal spaces assocated wth the MOD natural neutrosophc subset semgroup {S( Z ), +} under +, vz; T o, T and T are dstnct. Thus f S = {S( Z n ), +,,, } be a MOD natural neutrosophc subset under operatons or or + or. Then assocated wth S = S we get the followng MOD natural neutrosophc specal type of subset topologcal spaces of all types usng the MOD natural neutrosophc subset semgroup on {S, } or {S, } or {S, +} or {S, }. Let T o = {S,, }, T = {S, +, }, T = {S,, }, T = {S, +, T = {S,, } and T = {S, +, } be the sx dstnct MOD natural neutrosophc specal type of subset

23 MOD Natural Neutrosophc Subset Topologcal topologcal spaces assocated wth semgroups under the operatons + or or or respectvely. We wll llustrate ths by some examples. Example.11. Let S = {S( Z ), +} (or {S( Z ), } or {S( Z ), 15 } or {S( Z 15 ), }) be the MOD natural neutrosophc subset semgroups under + or or or respectvely. Let S = {S {}}; T o = {S,, }, T = {S, T = {S, T = {S,, }, T = {S, } and T = {S, +, } be the sx MOD natural neutrosophc specal type of subset topologcal spaces assocated wth the MOD natural neutrosophc subset semgroup usng S( Z ). 15 We show all the four operatons are dstnct hence the related topologcal spaces are also dstnct. and B = { 15 5 Let A = { ,, 1,, +,,, 9, 1,,, 1, , + 8, 15, } S , 11} A ,, 1,, 8, 15, , 11} {,,, 9, 1,,, 1, , 15, 15 } = { ,,, 1, , 11, , ,, 9, 1,, 15, 15 5 } Next we fnd A B = { 15 +, 11} { {,, } ,, 1,, 8 +, 15, 15,,, 9, 1,,, 1, A + B = { ,, 1,, 8 +, 1 + { ,,, 9, 1, 1,, 1, , 15, 15 5 } 15, } = , 11} +

24 MOD Subset Semgroups Under = { , , , , , , , , , 1, 9, , , ,, , 5, 1,, 11 +, , , , 11,, 1, + 15, , , 5,, 1,, , , 11, 5 + +, , , 1, 7,, , , , 8, ,, , , , 1, +, , , 1 +,, 1 + +, , + 15, } Now we fnd A B = { 15 +,,, 9, 1,,, 1, ,, 1,, 8 +, 15, , } , 11} { + = { , , , , , , , , , 1,,, , ,, , 9, ,, 1, ,, + 15, ,, 5, , 7, ,1, , , , , , + 15, , , , 15, , , 15, } Clearly all the four equatons are dstnct and hence all the sx MOD topologcal spaces are dstnct. Next we proceed onto defne the substructures lke MOD topologcal subset subsemgroups, MOD topologcal strong subset deals and MOD topologcal subset deals. Defnton.1. Let {S, (where S = (S( Z ) }) (or {S,, +} or {S, } or {S,, +} or {S,, }) be the MOD specal type of topologcal subset spaces n V

25 MOD Natural Neutrosophc Subset Topologcal A non empty subset {P, } (P S )(or {P, +} or {P, } or {P, }) s defned as a MOD subset topologcal subsemgroup space f {P, }, {P, +}, {P, } and {P, } are subsemgroups. We wll frst llustrate ths stuaton by some examples. Example.1. Let S = {{S( Z ), X} or or or +} be the MOD natural neutrosophc subset semgroup under {or or or +). Let S = S {} and T o = {S,, }, T = {S, +, T = {S, +, }, T = {S,, }, T = {S,, } and T = {S, +, } be the MOD natural neutrosophc specal type of subset topologcal spaces assocated wth any two of the MOD natural neutrosophc subset semgroups. Consder P o = {S(Z ) {},, } T o s a MOD subset natural neutrosophc specal type of topologcal subspace of T o. Ths s a very well known fact. But M = {, {, 5,, 8, 9}, {, 19, 18, 1, 5 }, } T o s a MOD natural neutrosophc specal type of subset topologcal space subsemgroup under. Clearly M s not a MOD natural neutrosophc specal type of subset topologcal space subsemgroup under or or +. Let L = {A = {, + 5, 1, 18 1,, 19}, B = {, + 1, + + = {, , 1, 1 + 1, 1, }, C + 5, 1, , 1,, 1, 19,, +1, }, } T o be a MOD natural neutrosophc + 15, 1, specal type of subset topologcal space subsemgroup under and not a MOD natural neutrosophc specal type of subset topologcal space subsemgroup under or + or.

26 MOD Subset Semgroups Under 5 Consder B = {A = {, 5,, 1,, }, C = {, 1,,, 1,,, }, D = {,, 1, 5 1, 1 }} T s a MOD natural neutrosophc specal type of subset topologcal subsemgroup under, clearly B s not a MOD natural neutrosophc specal type subset topologcal space subsemgroup under + or or. Thus these MOD natural neutrosophc specal type of subset topologcal spaces subsemgroup under each of the operatons. Now havng seen examples of MOD natural neutrosophc specal type of subset topologcal space subsemgroups we now proceed onto defne MOD natural neutrosophc specal type of subset topologcal space strong subsemgroup n the followng. Defnton.. 1 Let S = {S( Z )} be the MOD natural neutrosophc subset collecton. Let S = S {}, then {S, (or {S, } or {S, +} or {S, }) are MOD natural neutrosophc subset semgroups. n T o = {S,, }, T = {S,, }, T = {S,, +}, T = {S,, +}, T = {S,, } and T = {S, +, } be the MOD natural neutrosophc specal type of subset topologcal spaces P S be a MOD natural neutrosophc specal type of subset topologcal space subsemgroup under +,, and then we defne P to be the MOD natural neutrosophc specal type of subset topologcal strong subsemgroup. We wll gve frst example of ths stuaton. Example.1. Let S = {S( Z 1 ) be the MOD natural neutrosophc subset collecton. 5

27 MOD Natural Neutrosophc Subset Topologcal Let S = S {} and T o, T, T, T, T and T be the MOD natural neutrosophc specal type of subset topologcal spaces. Let B = {S(Z 1 ) {}} = {collecton of al subsets of Z 1 wth } = P(Z 1 ) S. Clearly B s a MOD natural neutrosophc specal type of subset topologcal strong subsemgroup of T o, T, T, T, T and T., 1 Let W = {A = { +,, 1, 1, , 1 +, }, {, 1 }= C, B = {, 1 1, }} S. W. Clearly W + W = W, W W = W W = W. W W = So W s a MOD natural neutrosophc subset specal type of topologcal subspaces of T o, T, T, T, T and T. W s clearly MOD natural neutrosophc specal type of topologcal spaces strong subsemgroups under +,, and. n vew of all these we have the followng theorem. Theorem.. Let S = {S ( Z ) ; n < } be the collecton of n MOD natural neutrosophc subsets of Z n. Let S = S {}. T o = {S,, }, T = {S,, }, T = {S,, +}, T = {S, T = {S,, } and T = {S, } be the MOD natural neutrosophc specal type of subset topologcal spaces. W S s a MOD natural neutrosophc specal type of subset topologcal space strong subsemgroup f and only f W s

28 MOD Subset Semgroups Under 7 a MOD natural neutrosophc specal type of subset topologcal subspaces of T o, T, T, T, T and T. reader. Proof s drect and hence left as an exercse to the However we supply an hnt to the proof of the theorem. f W S s a MOD natural neutrosophc specal type ofstrong topologcal space strong subsemgroup then by defnton we know {W, +}, {W, }, {W, } and {W, } are all MOD natural neutrosophc subset specal topologcal space subsemgroups. So {W,, }, {W, +, }, {W, +, }, {W,, }, {W,, +} and {W,, }are all MOD natural neutrosophc specal type of topologcal subset subspaces of T o, T, T, T, T and T respectvely. The converse follows from the smlar argument. Next we proceed onto descrbe and defne MOD natural neutrosophc specal type of ¾ strong topologcal spaces. Defnton.. Let S = {S( Z n )} be the MOD natural neutrosophc subset collecton. S = S {}. T o = {S,, }, T = {S,, +}, T = {S,, +}, T = {S, T = {S,, } and T = {S +, } be the sx dstnct MOD natural neutrosophc specal type of subset topologcal spaces. Let P S, f P s a MOD natural neutrosophc subset subsemgroup under any three of the operatons {+, and } or {, and } or {+, and } or {, + and. then we defne P to be the MOD natural neutrosophc specal type of ¾ strong subsemgroups subset topologcal space. We wll llustrate ths stuaton by some examples.

29 8 MOD Natural Neutrosophc Subset Topologcal Example.1. Let S = S( Z ) be the MOD natural neutrosophc subset. Let S = S {} be the set wth empty set. T o = {S,, T = {S, +, }, T = {S,+, T = {S,, }, T = {S,, } and T = {S,, +} be the MOD natural neutrosophc subset specal type of topologcal spaces related wth the MOD natural neutrosophc subset semgroups {S, +}, {S, }, {S, } and {S, }. {, }}. Let W = {{ ,, }, { + 1,, }, {W, } s a MOD natural neutrosophc specal type of topologcal space subset subsemgroup. {W, } s a MOD natural neutrosophc specal type of topologcal space subset subsemgroup. {W, +} s a MOD natural neutrosophc specal type of topologcal space subset subsemgroup. However {W, } s not a MOD natural neutrosophc subset specal type of topologcal space subsemgroup. Hence W s only a MOD natural neutrosophc specal type of ¾ strong topologcal space subset subsemgroup of S. +, Let P = {, + 1,, +,, 1, 1 1, + 1, 1 +, , ,, , , , } S be the MOD natural neutrosophc subset of S. {P, +}, {P, } and {P, } are MOD natural neutrosophc specal type topologcal space subset ¾ strong subsemgroup of S. Clearly P and W are dfferent.

30 MOD Subset Semgroups Under 9 nterested reader can fnd other two types of MOD natural neutrosophc specal type of ¾ strong topologcal space subsemgroups. Now we see W s a MOD natural neutrosophc subset specal type of topologcal subspaces of T, T and T assocated wth the MOD natural neutrosophc subset specal type ¾ strong topologcal space subsemgroup. n vew of all these we have the followng theorem. Theorem.. Let S = (S( Z n )) be the MOD natural neutrosophc subset collecton, n <. Let S = S {}; T o = (S,, ), T = {S, +, }, T = {S +, }, T = {S,, }, T = {S,, } and T = {S, +, } be the MOD natural neutrosophc specal type of subset topologcal spaces assocated wth the MOD natural neutrosophc subset semgroups {S, }, {S, }, {S,+} and {S, }. W S s a MOD natural neutrosophc subset specal type of ¾ strong topologcal space subsemgroup under any of the three trples bnary operatons {,, +} (or {or {, +, }, or {,, }) f and only f W s a MOD natural neutrosophc specal type of subset topologcal subspaces such as {W,, }, {W,, +} and {W,, +} f the operatons (,, +} s taken (lkewse for other trples). reader. Proof s drect and hence left as an exercse to the Next we proceed onto defne the noton of MOD natural neutrosophc specal type of subset ½ strong topologcal space subsemgroups under only any of the par of operatons {, } (or {, +} or {, } or {, } or {, +} or {+,}). We wll frst llustrate ths stuaton by some examples.

31 MOD Natural Neutrosophc Subset Topologcal Example.15. Let S = {S( Z )} be the MOD natural neutrosophc subset collecton. S = S S,, }, T = {S +, }, T = {S, +, }, T = {S,, }, T = {S,, } and T = {S,, +} be the MOD natural neutrosophc specal type of subset topologcal spaces assocated wth the MOD natural neutrosophc subset semgroups {S }, {S, }, {S,+} and {S, }. Let V S, f V s a MOD subset natural neutrosophc semgroup under any two of the operatons then V s a MOD natural neutrosophc subset specal type of ½ strong topologcal space subsemgroups. Take V = {{ +, 5 +, +, }} S. Clearly V s a MOD natural neutrosophc subset specal type of ½ strong topologcal space subsemgroups gven by {V, ) and {V, }. However {V, +} and {V, } are not MOD natural neutrosophc subset subsemgroups. Let W = {A = { +, + +, }, B = {, + +, }}; W s a MOD natural neutrosophc specal type of subset ½ strong topologcal space subsemgroup as {W, +} and {W, } are MOD natural neutrosophc subset subsemgroups of {S,+} and {S, } respectvely. Clearly W s not a MOD natural neutrosophc subset subsemgroup under or. n vew of ths we have the followng theorem. Theorem.. Let S = {S( Z )} be the MOD natural neutrosophc subset collecton. S = S {}; let T o = {S,, }, T = {S, +, }, T = {S, +, }, T = {S,, }, T = {S,,} and T n = {S, +, } be the MOD natural neutrosophc

32 MOD Subset Semgroups Under 1 specal type of subset topologcal spaces. V S, be a MOD natural neutrosophc specal type of ½ strong subset topologcal subsemgroup f and only f V s a MOD natural neutrosophc specal type of subset topologcal subspace of T o (or T or T or T or T or T ; or used n the naturally exclusve sense. reader. Proof s drect and hence left as an exercse to the We have seen examples of all types of MOD natural neutrosophc specal type of subset topologcal spaces. We next defne topologcally strong deals of MOD natural neutrosophc subset specal type of topologcal subspaces of T o, T, T, T, T and T. Defnton.. Let S = (S( Z n )) be the MOD natural neutrosophc subsets collecton S ={} S be the power set of Z. T o = {S,, }, T = {S, }, T = {S, }, T = n {S,, }, T = {S,, } and T = {S, +, } be the sx MOD natural neutrosophc specal type of topologcal subset spaces usng the MOD natural neutrosophc subset semgroups {S, +}, {S, }, {S, } and {S, }. Let W S be the MOD natural neutrosophc specal type of subset strong topologcal subspacesof MOD natural neutrosophc subset semgroups. ) W s a MOD natural neutrosophc deal of the MOD natural neutrosophc specal type of subset topologcal spaces f W s an deal wth respect all four products. Ths s possble only when W = S. ) W s a MOD natural neutrosophc ¾ super strong deal f W s a strong subset semgroup and W s an deal, only under any of the three

33 MOD Natural Neutrosophc Subset Topologcal ) v) products from,, + and ths happens only when W = S. W s a MOD natural neutrosophc super ½ strong deal f W s a strong subset semgroup and W s an deal when under any two of the products,,+ and. W s a MOD natural neutrosophc just an deal f W s a strong or ¾ strong or ½ subsemgroup but W s only an deal n one of the products +,, and. We wll llustrate ths stuaton by some examples. Example.1. Let S = {S( Z )} be the MOD natural neutrosophc subset topologcal spaces usng S = S {};T o = {S,, }, T = {S, }, T = {S,, }, T = {S, }, T = {S,, } and T = {S,, }. Let W = {Collecton of all subsets from {,,, 9, 1, 1, 1, 1, 1, 1 }}, {W,, } = W o, W = {W, , +}, W = {W,, +}, {W, W, W = {W,, } and W = {W, +,,} are all MOD natural neutrosophc subset strong specal type of topologcal subspaces assocated wth the MOD natural neutrosophc subsemgroups {W, }, {W, }, {W, +} and {W, }. Cleaarly W o, W, W, W, W and W are all MOD natural neutrosophc subset specal type of topologcal subspace deals under and that s A {W o } W o. (A W ) W, A W W A W W

34 MOD Subset Semgroups Under A W W and A W W for all A S. Smlarly A W W, A W W, A W W, A W o W o, A W W, and A W W for all A S. However A W o W o n general for all A S. A W o W o. For take A = {5, 7,, 11} and B = {, , + 1, }. A B = {5, 7,, 11} 1 9, + 1, } = {, 5, 7, 11,, , + 1,} W o. Hence the clam. Consder for the same A and B; A + B ={5, 7,, 11} + {, , + 1, } = {8, + 1 9, , 5, 1, + 1 9,, 11,, 1 + 1, 7, , 1 + 1, , } W o. However A B = {5, 7,, 11} {, , + 1, } = W o. A B = {5, 7,, 11} {, , + 1, } = {,, ,, 9, + 1, 1, 1, 9} W o Thus A W o W o for all A S and A W o W o for all A S. Hence W o s a MOD natural neutrosophc specal type of topologcal subset subspace deal only under and. Consder W the MOD natural neutrosophc specal type of topologcal subset subspace of S.

35 MOD Natural Neutrosophc Subset Topologcal W Now we see s also only a MOD natural neutrosophc specal type of topologcal subset subspace deal under and. A = {5 + B = { + 1,, } W { + 1, 7,, 9, 1, 8 + 1,, 1} S. Let 1. We frst fnd A + B ={5+ 1, 7,, 9, 1, 8 + 1,, 1} },, = { , 1 + 1, 5 + 1, 1, + 1, , 7, 1 + 1, 5 + 1, 7,, 9, 1, 8 + 1,, 1, , , , , , 5 + +, , } W so n general A + W cannot be a MOD natural neutrosophc specal W So W type of topologcal subspace subset deal of S under +. Consder for the same A S and B W. A B = {5 + 1, 7,, 9, 1, 8 + 1,, 1} { + 1,, } = { , 7,, 9, 1, 8 +,, 1, +,, } W. Thus W s not a MOD natural neutrosophc specal type of topologcal subset subspace deal of S under. Consder for the same A n S and n W, A + B = {5 +, 7,, 9, 1, 8 + 1,, 1} + { + 1,, } = { , 1 + 1, 5 + 1, 1 + 1, + 1, , 5 + 1, 7,, 9, 1, 8 + 1,, 1, , + 1 +, , , , , , } W

36 MOD Subset Semgroups Under 5 Thus we see W s not a MOD natural specal type of topologcal subspace subsets deal under +. n vew of ths we prove for S = {S( Z )} and S = S {}, t s mpossble to have MOD natural neutrosophc specal type of subset topologcal subspaces whch can be super deals. Hence we conjecture does there exst any specal type of topologcal space usng a unversal power set or otherwse say P such that {P,, }, {P, +, }, {P,, }, {P, +, }, {P, +, } and {P, } the sx dstnct topologcal spaces to have a nontrval proper subset M such that {M,, }, {M,, +}, {M,, }, {M,, +}, {M,, } and {M, +, } are subspaces and M s to be a super deal. Clearly we cannot have ths usng the power set of P( Z ); n <. n Now we proceed onto bult MOD natural neutrosophc specal type of topologcal subset spaces usng P(C( Z n )), P(Z n g ), P(Z n h ), P(Z n ) and P(Z n k ) and study the specal features assocated wth them by approprate examples. Example.17. Let V = P(C (Z 1 )) be the MOD natural neutrosophc fnte complex number subsets, that s power set of C (Z 1 ) = {a + b F + 1 t / a, b Z 1 ; F = 9 and t s a zero dvsor or dempotent or nlpotent n C(Z 1 )}. n {V,, } = V o s the MOD natural neutrosophc specal type of ordnary topologcal subset space of fnte complex numbers. Clearly M = P (Z 1 ) and N = P(C(Z 1 )) are MOD natural neutrosophc fnte complex number subset specal type of topologcal subspace of V.

37 MOD Natural Neutrosophc Subset Topologcal M o, M, M, M, M and M are MOD natural neutrosophc fnte complex number subset specal type of topologcal subspaces of M. Smlarly N o, N, N, N, N and N are MOD natural neutrosophc fnte complex number subset specal type of topologcal subspaces of M. Thus both M and N are MOD natural neutrosophc fnte complex number specal type of subset strong topologcal subsemgroups of V. }, {9,1, 5 + C 8 C C 8, C Consder B = {{ F, 9, C C, + C 5 + C }, {9, + + C C + + C C 8 C + C, + + C C, + C 5 C C C + C C C 8 C 8, + C C, F + }, {1, + C 8 + }, { C + C + C + C + C 8 + }, }. C 8 + C C + C C C +, 9 F, F + C C + C C + Fnd B,, the MOD natural neutrosophc fnte complex number subset subsemgroup. Then can we say {B,, } s a MOD natural neutrosophc fnte complex number subset subsemgroup? Let D = {{ + C,, C + C + C 8 + C + C } be a subset of V and {D, }, {D, } and {D, } are MOD natural neutrosophc fnte complex number ¾ strong subset subsemgroup. Hence, D o = {D,, }. D = {D,, } and D = {D,, } are all three MOD natural neutrosophc fnte complex number specal type of topologcal subset subspaces of V.

38 MOD Subset Semgroups Under 7 + {, + C, + C 8 + C Consder D + D = { + C,, } = { + + C C + C C C C C } D. Thus D s not closed under +. C + +,, + C + C } C + C C 8 } V. Let L = {5,, 5 + C, C, 5 + C 5, 5 + C + C 5, C 5 + C Clearly L + L = L, L L = L, L L = L and L L = L. Thus L s a MOD natural neutrosophc fnte complex number strong subsemgroup. L o = {L,, }, L = {L, +, }, L = {L,, }, L = {L, +, }, L = {L,, } and L = {L, +, } are MOD natural neutrosophc fnte complex number specal type of subset topologcal subsapces of V. We see none of them are deals. Take A ={5, } V, A L = {5} L. A L = {5, } L L. A + L = {5, } + L = {, 5, 8, 8 + C, + C, } L. A L = {, 5} L = {5,, 5 + C, C C 5, C 5, 5 + C + C 5, C 5 + C } = L.

39 8 MOD Natural Neutrosophc Subset Topologcal So only under wth ths A, t s closed. Let B = { C, C } V, B L = { C C C C, +,, C, C C + } B, so for ths B, B L B. Thus L cannot be an deal more so a super strong deal of V. n some cases even just an deal of V. Hence t s left as an open conjecture to fnd condton on subsets W S to be ) just an deal. ) ½ strong deal. (for ¾ strong deal and strong deal stuatons are ruled out). The exstence of ¾ strong deals and strong deals can be studed n case of other topologcal spaces. The strkng feature about these MOD natural neutrosophc fnte complex modulo nteger specal type of topologcal spaces of subsets usng P(C (Z n )) = S (C (Z n )) {} can be analysed by any nterested researcher. below. However some facts about ths structure s mentoned Let M = P(C (Z n )) be the MOD natural neutrosophc fnte complex number subset collecton. M o = {M,, }, M = {M, +, }, M = {M, +, }, M = {M, +, }, M = {M,, } and M = {M,, } are MOD natural neutrosophc fnte complex number specal type of topologcal subset space usng subsets of P(C (Z n )). We lst out the propertes enjoyed by these sx spaces.

40 MOD Subset Semgroups Under 9 Property.1. S = {P(Z n )) M s always a subspace of all the sx MOD natural neutrosophc fnte complex number specal type of topologcal subset spaces, M o, M, M, M, M and M. Property.. R = {P(C(Z n ))} M s always a subspace of all the sx MOD natural neutrosophc fnte complex number specal type of topologcal subset spaces M o, M, M, M M and M. Property.. W = P(Z n F ) where Z n F = {a F / a Z n }} M s only a subspace of M o, M and M. Clearly W s not a subspace of M, M and M as {W, } s not even a closed set under. Property.. M has no strong deals but has strong subsemgroups. Next we proceed onto study the propertes assocated wth MOD natural neutrosophc-neutrosophc subset specal type of topologcal space usng P(Z n ) by examples. Example.18. W = {P(Z ) be the MOD natural neutrosophc-neutrosophc subset. W o = {W,, }, W = {W,, +}, W = {W,, +}, W = {W, W = {W,, } and W = {W, +, } be the MOD natural neutrosophc-neutrosophc specal type of subset topologcal space usng subset semgroups (W, +),{W, }, {W, } and {W, }. We see M = {P(Z )} s a MOD natural neutrosophcneutrosophc specal type of subset topologcal subspaces gven

41 MOD Natural Neutrosophc Subset Topologcal by M o = {M,, }, M = {M, +, }, M = {M,, } and = {M,, }. None can contrbute to deals. M nfact M s a strong subset subsemgroup of W. Consder N = P(Z )} W, N s also a MOD natural neutrosophc-neutrosophc specal type of subset topologcal subspaces; N o = {N,, }, N = {N,, +}, N = {N, +, } N = {N,, }, N = {N,, } and N = {N, +, }. N s only a MOD natural neutrosophc-neutrosophc strong subset subsemgroup and s not an deal. However M N, proper contanment. n vew of ths we can defne the noton of restrcted deals or restrcted quas vector spaces. We see N s an deal over M nfact a strong deal. Also N can be vewed as a quas specal vector space over M where M s not the feld whch the classcal defnton of vector space requres. Thus f L ={S({,, })} {}. Then L s a MOD natural neutrosophc-neutrosophc specal type of topologcal subset subspaces gven by L o = {L,, }, L = {L, +, }, L = {L,, +}, L = {L,, +}, L = {L,, } and L ={L,, }. nfact both M and N are MOD natural neutrosophcneutrosophc specal quas vector spaces over L. Smlarly f T = P({, }) W, then T s a MOD natural neutrosophc-neutrosophc specal type of subset topologcal subspaces assocated wth T gven by T o = {T,, }, T = {T,

42 MOD Subset Semgroups Under 1 +, }, T = {T, +, }, T = {T, +, }, T = {T,, } and T = {T, }. Clearly L, M and N are MOD natural neutrosophc-neutrosophc specal quas vector spaces over T. However T s a not a MOD natural neutrosophcneutrosophc specal quas vector space over L or M or N. Consder B = {P(Z ()} W, s agan a MOD natural neutrosophc-neutrosophc subset specal type topologcal subspace of W. Clearly B s not an deal. Let D = {P(Z t ) / t Z s an dempotent or nlpotent or a zero dvsor} W be a MOD natural neutrosophc-neutrosophc specal type of subset topologcal subspace assocated wth strong subset semgroup D. Clearly D s not a strong deal of W. However D s a ¾ strong deal of W. A + W W n general for A W. We have followng propertes assocated wth MOD natural neutrosophc-neutrosophc specal type of topologcal subset spaces assocated wth the subset semgroups bult usng W = P(Z n ), W o = {W,, }, W ={W, +, }, W = {W, +, }, W = {W,+, }, W = {W,, } and W ={W,, }. Property.5. Let W = {P(Z n )} be the sx topologcal spaces mentoned above. W has MOD natural neutrosophcneutrosophc specal type of subset strong subsemgroup P. Assocated wth P we have the sx MOD natural neutrosophc-

43 MOD Natural Neutrosophc Subset Topologcal neutrosophc specal type of subset topologcal subspaces, P o, P, P, P, P and P. Property.. W has no MOD natural neutrosophc-neutrosophc strong deals. Property.7. Let W = (PZ n ). All the assocated MOD natural neutrosophc specal type of subset topologcal spaces has MOD topologcal subspaces B whch are MOD quas specal vector spaces over certan other subspaces C such that C B. Property.8. E = {P(Z n t ) / t s an dempotent or nlpotent or a zero dvsor n Z n } W s such that E s ½ strong deal of W. Next we proceed onto descrbe by examples MOD natural neutrosophc dual number specal type of topologcal subset spaces usng P(Z n g ). Example.19. Let Z = {P(Z 1 g )} be the MOD natural neutrosophc dual number specal type of subset topologcal space assocated wth MOD natural neutrosophc dual number subset semgroup {Z, }, {Z, }, {Z, +} and {Z, }. Let Z o = {Z,, }, Z = {Z, +, }, Z = {Z, Z = {Z, +, }, Z ={Z,, } and Z = {Z, } be the MOD natural neutrosophc dual number subset specal type of topologcal subset spaces assocated wth Z. Let A = {+ g, 8 +g, 5 +g, B = {g +, 5g, g, 9g, g g, +g} Z. g, g g, g, g, 11g} and We wll fnd A + B, A B, A B and A B.

44 MOD Subset Semgroups Under A + B = { + g, 8 + g, 5 + g, g, g g, g, g, 11g} + {g +, 5g, g, 9g, g g, + g} = {8 + g, + 5g, 9 + 7g, g + + g, + g + g g, + g, g +, g +, + 5g, 8 + 8g, 5 + 9g, 5g + g, g, g, 5g + g + g, g + 9g, 5 + g, g g + g g, 7g, 9g, g, + g + g g + g, g + g, g + g g + 9g, 11g, g, 8g, g, g g + g g, g, 8 +g + g, 11g + g g + + g, g g + g, g + g, 5 + g + g, +1g, 8, 5 + g g g, g g, 11g + g g + g g, 7 + g, g g g, 8 + g, + + g, + g, g + + g, + g, + g} A B ={ + g, 8 +g, 5 + g, 5g + g, 9g, g, g g, g, g, 11g} {g +, g g, g, g, 11g} = { + g, 8 + g, 5 + g, g, g, 11g, g +, 5g, g, g g, 9g, g, g, 11g} g, g g, A B = { + g, 8 + g, 5 + g, g, 5g, g, 9g, g g, + g} = A B = { + g, 8 + g, g +5, 5g, g, 9g, g,, g g, g, g g, + g} = {, g + 8, 8 + g, g g, g, g, 11g} {g +, g g, g, g, 11g} {g +, g, g g, 8g, g, g, g g, 11g, + 1g, g, 9g} V Clearly the equatons,, and V are dfferent hence all the sx MOD natural neutrosophc dual number subset specal topologcal spaces Z o = {Z,, }, Z = {Z, +, }, Z = {Z, +, }, Z = {Z,, }, Z = {Z, +, } and Z = {Z,, } all are dstnct. One can have subspaces assocated wth all these spaces.

45 MOD Natural Neutrosophc Subset Topologcal For nstance W = (P(Z 1 ) Z = P(Z 1 g )} Z s a proper subset of Z and W o = {W,, W = {W, +, }, W = {W, +, }, W = {W, +, }, W = {W,, } and W = {W,, } are all MOD subset specal type of topologcal subspaces of Z. However W s not an deal of Z. Consder V ={P(Z 1 g)} Z. We see V s a MOD specal type of subset topologcal subspace; V o ={V,, }, V = {V, +, }, V = {V, +, }, V ={V, +, }, V ={V,, } and V = {V,, } are all dstnct MOD natural neutrosophc subset specal type of topologcal subspaces of Z and none of them are deals of Z. g Consder B ={P(Z 1 g t ) where t Z 1 g s a zero dvsor or an dempotent or a nlpotent element of Z 1 g)} Z. Clearly B s a MOD natural neutrosophc dual number subset specal type of topologcal subspaces gven by B o = {B,, }, B = {B, +, }, B = {B, +, }, B = {B,, }, B = {B, +, } and B = {B,, }. Clearly B s an not deal under all four operatons. B s an deal only under and operatons. Clearly under and + they are not deals. nterested reader can prove that there s no nontrval MOD subset natural neutrosophc dual number specal type of topologcal subspace whch s an deal under + and. For t can be easly establshed.

46 MOD Subset Semgroups Under 5 Next we proceed onto descrbe by an example the MOD natural neutrosophc specal dual lke number specal type of topologcal subset space usng P(Z n h ) where Z n h ={a + bh / a, b Z n, h = h} and Z n h = {Collecton of all MOD natural neutrosophc specal dual lke numbers} = {a + bh + h t / a, b Z n and t s a nlpotent or a zero dvsor or an dempotent of Z n h}; n <. Example.. Let S = {P(Z 1 h } be the MOD natural neutrosoophc specal dual lke number subsets collecton ncludng. {S, }, {S, }, {S, } and {S, +} are all MOD natural neutrosophc specal dual lke number subset semgroups. Hence we obtan the correspondng sx MOD natural neutrosophc specal dual lke number specal type of subset topologcal spaces. S o = {S,, }, S = {S, +, }, S = {S, +, }, S = {S, +, }, S = {S,, } and S ={S,, }. All the sx spaces are dstnct evdent from the followng. Let A = { + 5h, h, 9h + 1, + h, 5 + h, h 5 } and B = { h 5, h, + h, h + h + h, h + h} S. We fnd A B, A B, A + B and A B. A B = { + 5h, h, 9h + 1, + h, h + + h, h + h} h, h + h} = { + 5h, h, 9h + 1, + h, 5 + h, h, 5 + h, h 5 } { h 5, h, + h 5, h +, + +

47 MOD Natural Neutrosophc Subset Topologcal A B = { + 5h, h, 9h +1, + h, 5 + h, h + h + h, h + h} = { h 5, h} h, h 5 } { h 5, h, + A + B ={ + 5h, h, 9h + 1, + + h, + h + h, h + h} h, h 5, 5 + h } + { h 5, h, = { + 5h + h h 5, h + 5 1, + h + h h h 5, h + 5, h, + h +, 9h h, + h 5, + h + h +, 5h + h h 5, h h 5, 5 + +, + 7h, h + h, h + 5h + 5, + 7h, 5 + h + 5h, 5 + 9h, + h + h, 5h + h, 9h h 5, h + 1, + 9h + h, h + + h, h + h +1, + h + h h 5, + h +, + h + h, + h + h h +, h h + h +, h + 5, + h h 5 +, + h + h h h h + + h 5 h h h h h, , 5 + h +, 7 + +, 7 + h + h h +, 5 +h + h + h } We fnd A B ={ + 5h, h, 9h + 1, + h, + h, h + + h, h + h} = { h 5, h 5 +, 5h +, h, h,, h + h h h + h, + h + h } h + h, h 5, h h, h, + h, h + h h h, + 8h + +, h + h + 5} { h 5,, + 8h + h, h h h + 5, V Clearly the equatons,, and V are dstnct hence S o = {S,, }, S = {S, +, }, S ={S, +, }, S ={S,, }, S = {S, +, } and S = {S,, } all are dstnct MOD natural neutrosophc specal dual lke number specal type of subset topologcal spaces bult usng {S, }, {S,,{S, +} and {S, } the four MOD natural neutrosophc specal dual lke number subset semgroup. Next we proceed onto descrbe MOD natural neutrosophc specal quas dual number subset specal type of

48 MOD Subset Semgroups Under 7 topologcal spaces usng P(Z n k ) where Z n k = {a + bk / a, b Z n, k = (n 1)k} and Z n k = {a + bk + k t / a, b Z n, t s an dempotent or nlpotent or zero dvsor of Z n k}. We wll descrbe ths stuaton by some examples. k Example.1. Let B = {P(Z k )} = {a + bk + t / a, b Z, k = k, t Z k = {a + bk / a, b Z } and t s an dempotent or a nlpotent or a zero dvsor}, be the collecton of MOD natural neutrosophc specal quas dual number subsets from Z k. We see {B, }, {B, }, {B, } and {B, +} are MOD natural neutrosophc specal quas dual number subset semgroups. We frst show all the four MOD natural neutrosophc quas dual number subset semgroups are dstnct. + k Take A = { + k, +, } and B = {, k, + k, k + 1, k k k k + k + k } B. k k k A B = { + k, + k, k k, + k k + k } {, k, +k, k +1, } = {, k} k k k A B = { +k, + } {, k, + k, k +1,, k k k + } = { + k, + k k k, k, + k k k, k + 1, k + k +, k, k, k + k k, + k k k,, k, 1 + k + k k k k k + +, k,, + +, k,, + k k k k k,, k, 1 + k + k k k k, k +,, k, 1 + k, k +, k,, k +,, k, 1 + k + k k + k + k } + k + k k + k + k + + k + k k, + k k

49 8 MOD Natural Neutrosophc Subset Topologcal A + B = { + k, + {, k, + k, k +1, {k,, + k k k k, 1 +, k k k k k, + k k k,, k, 1 + k + k k k + +, k,,,, + k, + k + k k k + k k k +, + k, k, 1 + k k k, + k + k k } + k k } = k, + k, + + k, k + k k k k, + k + k +, 1, 1 + k,, + k +, + k +, 1 + k, + k, k + +,, k + + k k k k k k k k,, k k, k + k k k k + k k k k k k k k k k k k, k +, k +1, + k + + k k k k k k, k + + k, k +, k + + k k k +, k + 1 +, + + k +, + k + +,, + k +, k + k k, + k + k, + k, k, 1 + k + + k k k, + k +, + k k k k k k k k + k, + k +, +, k +, k + k, + + +, + k + + +, + + k +, + + k + k k k + k k k + k k k + k, 1 + k + k + k + k k + We next fnd A B. A B = { + k, + k k k, +k + k k k + k + k k k k k k, k + },, + k k k k k k k + k k,, k, 1 + k +, k + k + k k k k k + k k } {, k, + k, k + 1, + k +, k, k + k k } = {, k, k k k, + k, k, + k + k k + k, k + k, k k, k k + k, k, k + k k + k, + k, + k + k k k, k, 1 + k + k, k k k k k k k + k + + k, +, + k + k,, 1 + k + + k, k + k k k k k k k, + k + k, + k +, k + + k k k, + k + k + k + k, k, k + k + k k k, k + k + k k k + k } V All the four equatons are dstnct hence the MOD subset natural neutrosophc specal quas dual number semgroups assocated wth them are also dstnct. Now we proceed onto descrbe the new specal type of topologcal spaces assocated wth them.

50 MOD Subset Semgroups Under 9 B o = {B,, }, B = {B, +, B = {B, +, }, B = {B,, }, B = {B,, } and B = {B,, +} are the sx dstnct MOD natural neutrosophc specal quas dual number subset specal type of topologcal space assocated wth the MOD natural neutrosophc subset specal quas dual number subset semgroups, {S, }, {S, }, {S, +} and {S, }. t s pertnent to keep on record that none of these MOD natural neutrosophc subset specal type of topologcal spaces have MOD strong subspaces whch are not MOD strong deals. They can never be a strong deal, to be more precse they cannot be deals under + and, they can be deals only under and. n all cases they are deals under but only n few cases they are deals under. nterested reader can analyse ths stuaton. Next we proceed onto construct MOD matrx subset specal type of topologcal spaces usng subsets of a MOD matrx semgroups under or or + or. We wll frst llustrate ths stuaton by some examples. a a Example.: Let M = { a a a 1 a a 5 a 7 8 / a Z 1, 1 < 8} be the collecton of all MOD matrces. Let R = P(M) = {collecton of all subsets of M}. {P(M), } be a MOD subset matrx semgroup under assocated wth M.

51 5 MOD Natural Neutrosophc Subset Topologcal {P(M), } s agan a MOD matrx subset semgroup under assocated wth M. {P(M), n } s a MOD matrx subset semgroup under n {P(M), +} s a MOD matrx subset semgroup under +. We shall show all the four MOD matrx subset semgroups are dstnct. 9 Let A = {, 1 9 A + B = {, } and B = { } P(M) } + { } = {, A B = {, 1 5 } } { }

52 MOD Subset Semgroups Under 51 9 = {, 1 1, } A B ={, }{ } = { } 9 A n B = {, 1 = {, } n { } } V 1 All the four equatons are dstnct, hence the four MOD subset matrx semgroups are dfferent. Now usng these four MOD subset matrx semgroups buld the sx dfferent MOD subset specal type of topologcal spaces. R o = {R,, } the MOD ordnary subset specal type of matrx topologcal space.

53 5 MOD Natural Neutrosophc Subset Topologcal R = {R, +, } s a MOD subset specal type of matrx topologcal space dfferent from R o. R = {R, +, } s also a MOD subset specal type of matrx topologcal space dfferent from R o and R. R n = {R, +, n } s a MOD subset specal type of matrx topologcal space dfferent from R o, R and R. n R = {R,, n} s a MOD subset specal type of matrx topologcal space dfferent from R o, R, R and R. n n R = {R,, n} s a MOD subset specal type of matrx topologcal space dfferent from R o, R, R, R n and R. n Thus t s a matter of routne to prove all the sx MOD subset specal type of topologcal spaces are dfferent n ther propertes. Example.. Let W = {(a 1, a, a, a ) / a Z 15 ; 1 } be the collecton of all MOD row matrces wth entres from Z 15. Let S = P(W) = {collecton of all matrces subsets from W} be the collecton of all subsets of W ncludng the empty set. S o ={S,, }, S = {S, +, }, S = {S, +, }, S = {S, +, }, S = {S,, } and S = {S,, } are all sx dstnct MOD subset matrx specal type of topologcal spaces. Clearly S o, S, S, S, S and S have MOD subset specal type of strong matrx topologcal subspaces whch are not strong deals.

54 MOD Subset Semgroups Under 5 Ths work s also a matter of routne so left as an exercse to the reader. Now we proceed onto show we can have MOD subset specal type of matrx topologcal spaces whch can be non commutatve. Such type s possble only n case of MOD matrx subset topologcal spaces of specal type. We wll llustrate ths stuaton by an example or two. a1 a Example.. Let B = { a a / a Z ; 1 } be the MOD square matrces. Let P = S(B) ={collecton of all matrx subsets from B} be the MOD matrx subset collecton ncludng the empty set. P o = {P,, }, P = {P, +, }, P ={P, +, }, P = {P, +, }, P = {P,, } and P = {P,, } be the sx dstnct MOD matrx subsets specal type of topologcal spaces usng P = S(B). Clearly P, P and P are the three MOD matrx subsets noncommutatve specal type of topologcal spaces. Let A ={ 1 5, 1, 1 } and B = { 1, 1 } P (or P or P ). Consder A B ={ 1 5, 1, 1 } { 1,

55 5 MOD Natural Neutrosophc Subset Topologcal 1 } = { 1 5 1, 1 1, 1 1, , 1 1 } = { 5, 5, 1,, 1, 1 } Consder B A = { 1, 1 } { 1 5, 1, 1 } = { 1 1 5, 1 1 5, 1 1, 1 1, 1 1, 1 1 } = { 1 5, 1 5 1,,,, 1 } Clearly and are dfferent, that s A B = B A. Hence we see of the sx MOD matrx subset specal type of topologcal spaces three are non commutatve and three are commutatve and three are commutatve.

56 MOD Subset Semgroups Under 55 The non commutatve topologcal spaces can occur only when the MOD matrces under consderaton are square matrces and the product s taken as the usual matrx product. However f s replaced by n, then the all the sx MOD subset matrx specal type of topologcal spaces would only be commutatve. One can get strong subspaces and subspaces and deals. All these are left as an exercse to the reader. Next we proceed onto descrbe MOD dual number subset matrx specal type of topologcal spaces by some examples. a1 a Example.5. Let W = { a a a 5 / a Z 1 g; 1 5} be the collecton of all 5 1 matrces. S(W) be the collecton of subsets of W. Let S(W) = D, {D, }, {D, }, {D, +} and {D, n } are four dstnct MOD subset semgroups. We see D o = {D,, }, D = {D, +, }, D = {D, +, }, D = {D,, }, D = {D,, } and D = {D, +, } are the MOD sx dstnct subset matrx topologcal spaces assocate wth D. Fndng MOD topologcal subspaces and deals happens to be a matter of routne so left as an exercse to the reader.

57 5 MOD Natural Neutrosophc Subset Topologcal a1 a Consder E = {subsets of matrx from P ={ a a a 5 / a Z 1 g; 1 5}} D s such that E o = {E, E = {E, +, }, E = {E, +, }, E ={E, +, }, E = {E,, } and E = {E,, } are MOD subset dual number specal type of topologcal subspaces and none of them are strong deals. They are all deals only wth respect to the two operatons and and not under the operatons + and. The specalty about ths E s such that E E = { }. Thus we call E the MOD dual number specal type of subset topologcal subspace whose product yelds zero subset matrx. Ths s the specalty enjoyed only by the MOD dual number matrx subset specal type of topologcal subspace of D. Next we proceed onto descrbe MOD fnte complex number matrx subset specal type of topologcal spaces by some examples.

58 MOD Subset Semgroups Under 57 a a a a a a a a a a a a Example.. Let S = { a a a a a a a a a a a a / a (Z 15 ) = {a + b F / a, b Z 15, F = 1}, 1 } be the MOD fnte complex number matrces. P(S) = V = {collecton of all subset matrces of S}. V s a MOD subset fnte complex number matrx semgroups under (or or or +). We can have the correspondng MOD fnte complex number subset matrx specal type of topologcal spaces vz., V o ={V,, V = {V, +, }, V = {V, +, }, V = {V, +, }, V = {V,, } and V = {V,, }. a a a a a a Consder P = { a a a a a a a a a a a a / a Z 15, F = {a F / a Z 15 }; 1 18]; S(P) = {collecton of all matrx subsets from P}. Let S(P) = L; L o = {L,, }, L = {L, +, } and L = {L, +, } are MOD fnte complex number subset matrx specal type of topologcal subspaces of L o, L and L respectvely. However under n nether P nor S(P) = L s even a closed set.

59 58 MOD Natural Neutrosophc Subset Topologcal L o, L and L are deals under only. That s f A V and B L o or L and L then A B L o (or L and L ) hence the clam. So here we can have the man feature assocated wth MOD fnte complex number subset matrces. Theorem.5. Let M = {s t matrces wth entres from C(Z n ); s t, s, t < }. V = P(M)={collecton of all subsets of M that s power set of M}. ) {V, }, {V, }, {V,+} and {V, n } are the four dstnct MOD fnte complex number subset matrx semgroups. ) V o = {V,, }, V = {V, +, n }, V = {V, +, ) n }, V n = {V, +, }, V = {V,, } and n V = {V,, n } are the sx dfferent MOD fnte complex number subset matrx specal type of topologcal spaces. f P(L) = {Power set of L where L = {s t matrces wth entres from Z n F, s t} M} = W, then W s only a MOD fnte complex number subset matrx topologcal subspaces W V, W V and W o V o and L as well as P(L) s not even closed under n. v) W o, W and W are deals only under. Proof s left as an exercse to the reader. Next we proceed onto descrbe MOD neutrosophc subset matrx specal type of topologcal spaces by some examples. a1 a a a Example.7. Let S = { / a Z ; 1 a5 a a7 a8

60 MOD Subset Semgroups Under 59 8} be the MOD neutrosophc matrx collecton wth entres from S. Let M = P(S) = {collecton of all subsets of S or power set of S} be the MOD neutrosophc subset matrx. We can have sx dstnct MOD neutrosophc matrx subset specal type of topologcal spaces gven by M o = {M,, }, M = {M, +, n }, {M, +, }, = M, M = {M, +, }, n n M = {M, n, } and fnte cardnalty. n M = {M, n, }. All of them are of Let N = P(T) = {collecton of all subset matrces from a1 a a a T = { a5 a a7 a8 where a Z ; 1 8}} M. Clearly N o = {N,, }, N = {N, +, n n }, N = {N, +, }, N = {N, +, }, n N = {N, n n, } and N = {N, n, } are the -dstnct MOD neutrosophc matrx subset specal type of topologcal subspaces of M o, M, M, M, n M and n respectvely. n M Clearly all the sx spaces are deals only under the operaton n and and not under + and. Thus MOD neutrosophc subset matrx specal type of topologcal spaces has no MOD neutrosophc subset matrx specal type of topologcal space deals only deals under n and. Next we proceed onto descrbe MOD specal dual lke number matrx subset specal type of topologcal spaces usng t s matrx collecton wth entres from Z n h = {a + bh / a, b Z n, h = h} by some examples.

61 MOD Natural Neutrosophc Subset Topologcal a Example.8. Let L ={ a a a 1 a a 5 / a Z 1 h; h = h; 1 } be the collecton of all matrces wth entres from Z 1 h. P(L) = W = {Power set of L that s collecton of all subsets from L ncludng the empty set}. Defne the four operatons +,. n, and on W so that {W, +},{W, n }, {W, } and {W, } are all MOD specal dual lke number subset matrx semgroups. All the four semgroups are dstnct. Consder W o = {W,, }, W = {W, +, n n }, W = {W, +, }, W n = {W, +, }, W = {W, n n, } and W = {W, n, } all the sx are dstnct MOD specal dual lke number subset matrx specal type of topologcal spaces assocated wth the four semgroup. deals. The reader s left wth the task of fndng subspaces and Let V = P(T) = {power set of T where a T = { a a a 1 a a 5 / a Z 1 h, 1 } W. }, V We see V o = {V,, }, = {V, +, }, n V = {V, +, n n }, V = {V, +, V = {V, n n, } and V = }V, n, }

62 MOD Subset Semgroups Under 1 are the sx dstnct MOD specal dual lke number subset matrx specal type of topologcal subspaces of W o, W, W, W, n W and n W respectvely. All of them are deals under n and however more of them s an deal under + and or (). So V cannot contrbute for a strong deal though V s a strong subsemgroup hence a strong MOD specal dual lke number matrx subset specal type of topologcal subspaces. nterested reader can study the specal features assocated wth them. Next we proceed onto descrbe by example the MOD specal quas dual number subset matrx specal type of topologcal spaces usng Z n k = {a + bk / a, b Z n, k = (n 1) k}. Example.9. Let S = P(W) = {Power set of W where n a1 a a W = { a a 5 a / a Z 1 k, k = 11k, 1 }} be the MOD specal quas dual number subset matrces wth entres from Z 1 k. {S, +}, {S, n }, {S, } and {S, } are MOD specal quas dual number subset matrx semgroups. All the four

63 MOD Natural Neutrosophc Subset Topologcal semgroups are dstnct. We see all the sx MOD specal quas dual number matrx subset specal type of topologcal spaces. all dstnct. S o = {S,, }, S n = {S, +, n n } and S = {S,, n} are Fndng deals, subspaces and strong subspaces s a matter of routne so left as an exercse to the reader. Next we proceed onto descrbe MOD natural neutrosophc subset matrx specal type of topologcal spaces n bult usng Z n = {a, b + t / a, b Z n, t s an dempotent or zero dvsor or nlpotent element of Z n ; that s elements n Z n whch are not nvertble n Z n by examples. Example.. Let V = P(S) = {Power set of S where S = {(a 1, a, a, a, a 5 ) / a Z ; 1 5} be the MOD natural neutrosophc subset matrx specal type of topologcal spaces assocated wth the MOD natural neutrosophc subset matrx semgroups {V, +},{V, n }, {V, } and {V, }. The related MOD natural neutrosophc subset matrx specal type of topologcal spaces are V o = {V,, }, V = {V, +, n }, V = {V,, +}, V n = {V, +, }, V = {V, n n, } and V = {V, n, }. We can fnd deals, subspaces and other substructures. Let A = {(,,,, ), (,,,, )} and B = {(,,,, ), (,,,, ), (,,,, )} V. We see A B = {(,,,,,), (,,,, )} {(,,,, ), (,,,, ), (,,,, )} = {(,,,, )}. Thus A and B s a MOD subset matrx topologcal zero dvsor. n

64 MOD Subset Semgroups Under Let A = {(,,,, ), (,,,, )} V, we see A A = {(,,,, )}; thus A s a MOD subset matrx topologcal nlpotent element of V. Let D = {(, 1, D D = {(, 1, = {(, 1,, 1, )} V, 1, )} {(, 1,, 1, )},1, )} s the MOD subset matrx topologcal dempotent of V. We defne the followng. Defnton.5 Let P(S) = V = {Power set of S where S = {s t matrces wth entres from Z n (or Z n or Z n g or Z n h or Z n k or C(Z n )) MOD subset matrces n, s, t <. V be the MOD subset matrx specal top ologcal spaces of dfferent types. We say A, B V s a MOD subset matrx specal type of topologcal zero dvsor par f A n B = {()} matrx. We say A V s a MOD subset matrx specal type of topologcal dempotent f A n A = A. We defne A V to be a MOD subset matrx specal type of topologcal nlpotent f A n A ={()}. We wll frst llustrate ths by some examples. Example.1. Let V = {(P(M), the collecton of all subsets

65 MOD Natural Neutrosophc Subset Topologcal a1 a from M where M = { a a a 5 / a Z 1, 1 5} be the MOD subset matrx collecton V o = {V,, }, V n = {V, +, n }, V = {V, +, }, V = {V, +, }, n V = {V, n n, } and V = {V, n, } be the MOD subset matrx specal type of topologcal spaces. Let A = {, 9, } and B = {, 8, 8 8, } V. We see A n B ={ }. Thus A, B n V s a MOD 8 subset matrx specal type of topologcal zero dvsor par. Let B ={ 1, , 9 9 } V. We fnd B n B; 1 1

66 MOD Subset Semgroups Under 5 B n B = { 1, , 9 9 } n { 1, , 9 9 } 1 1 = { n 1, n n, n n, n } ={ 1, , , 9 9 }. 1 1 Ths s the way product operaton n s performed on V. 9 9 The set { } = W n V s such that W n W = { 9, 9 9 9, }= W. 9 9

67 MOD Natural Neutrosophc Subset Topologcal Thus W s a MOD dempotent matrx subset. Smlarly f T = {,, } and S ={, } V then T n S ={ } s a MOD subset matrx zero dvsor. nterested reader can study MOD subset matrx dempotents, MOD subset matrx nlpotents and MOD subset matrx zero dvsors of the MOD subset matrx specal type of topologcal spaces. Ths study can also be extend to MOD natural neutrosophc subset matrx specal type of topologcal spaces bult usng P(M) where M s a n t matrx wth entres from Z n or C (Z n ) or Z n or Z n g or Z n h or Z n k. Such study s left as an exercse to the reader. Fndng MOD natural neutrosophc subset matrx specal type of topologcal zero dvsors, nlpotents and dempotents.

68 MOD Subset Semgroups Under 7 Further the study of MOD subset specal type of topologcal spaces usng W = S(P[x]) where P[x] = { a x / a Z n or C(Z n ) or Z n or Z n g or Z n h or Z n k or Zn or C (Z n ) or Z n or Z n g or Z n h or Z n h where {W, }, {W, }, {W, } and {W, +} are MOD subset polynomal semgroups and W o, W, W, W, W and W are MOD subset polynomal specal type of topologcal spaces. We wll provde one or two examples of them. All these MOD topologcal specal type of spaces are of nfnte order. Example.. Let S = {P(M[x]) where M[x] = { a x / a Z 1 g}} = {collecton of all subsets from M[x]}, {S, +}, {S,, {S, } and {S, } are all MOD subset dual number polynomals S o, S, S, S, S and S are MOD subset dual number polynomal specal type of topologcal spaces, bult usng the MOD subset dual number polynomal semgroups. Let A = {x + gx + 1, (5g + )x + g} and B = {gx + 1, (5g + )x + g} S. A B = {x + gx + 1, (5g + )x + g, gx + 1} A B = {(5g + )x + g} A + B = {gx + x + gx +, (5g + )x + gx + g + 1, (9g +)x + g x, (1g + )x + g} A B = {gx 5 + gx + x + gx + 1, gx + (5g + )x + g + gx (15g +9)x + 1gx + (5g +)x + gx + g, (g + 9)x + 1gx} V

69 8 MOD Natural Neutrosophc Subset Topologcal All the four equatons are dstnct so the four semgroups are dstnct hence the sx MOD specal type of subset polynomal dual number topologcal spaces are dstnct. Fndng MOD dual number subset topologcal spaces zero dvsors, nlpotents and dempotents happens to be a matter of routne. Example.. Let R = {P(T[x]) where T[x] = { a x / a Z 7 g }} = {collecton of all MOD natural neutrosophc dual number subset polynomal collecton). {R, }, {R, }, {R, } and {R, +} be the MOD subset natural neutrosophc dual number semgroups. Usng them we bult R o, R, R, R, R and R be the MOD natural neutrosophc dual number polynomal subset specal type of topologcal spaces. The reader s left as an exercse to fnd MOD natural neutrosophc dual number subset polynomal topologcal spaces zero dvsors, nlpotents and dempotents. Example.. Let W = {P(S[x])} = {collecton of all subsets polynomal from S[x] = { a x / a Z 1 }} be the MOD natural neutrosophc-neutrosophc subset polynomal collecton. W o, W, W, W, W and W are the MOD natural neutrosophc-neutrosophc subset specal type of topologcal polynomal spaces bult usng {W, +}, {W, }, {W, } and {W, } the MOD natural neutrosophc subset polynomal semgroups. Fndng MOD natural neutrosophc-neutrosophc subsets polynomal specal type of topologcal space dempotents, nlpotents and zero dvsors are left as exercse to the reader.

70 MOD Subset Semgroups Under 9 Example.5. Let S ={(P(M[x] 9 ) = {collecton of all subsets from M[x] 9 = { 9 a x / a Z 1 ; x 1 }} be the MOD subset fnte degree polynomal collecton {S, +}, {S, }, {S, } and {S, } be the MOD subset polynomal semgroups, S o, S,S, S, S and S be the MOD subset polynomal specal type of topologcal space. All propertes can be studed by the nterested reader. Example.. Let T = {S(P[x] 18 )} = {Collecton of all subsets from P[x] 18 = { 18 a x / a Z 19 h ; h = h, x 19 = 1 }} be the MOD natural neutrosophc specal dual lke number subset polynomals collecton T o, T, T, T, T and T be the MOD natural neutrosophc specal dual lke number subset polynomal specal type of topologcal spaces bult usng MOD natural neutrosophc specal dual lke number subset polynomal semgroups {T, }, {T, }, {T, } and {T, +}. Studyng the MOD natural neutrosophc specal dual lke number subset polynomal specal type of topologcal space zero dvsors, dempotents and nlpotents s consdered as a matter of routne and hence left as an exercse to the reader. Also order of these specal type of topologcal spaces are of fnte order. Example.7. Let B = {P(S[x] 5 } = {collecton of all subsets from S[x] 5 = { 5 a x / a C (Z 1 ); x = 1 } be the MOD natural neutrosophc fnte complex modulo coeffcent polynomal subset collecton} be the MOD natural neutrosophc fnte complex modulo nteger subset polynomal collecton. {B, +}, {B, }, {B, } and {B, } be the MOD natural neutrosophc fnte complex number subset polynomal semgroups.

71 7 MOD Natural Neutrosophc Subset Topologcal Let B o, B, B, B, B and B be the MOD natural neutrosophc fnte complex number subset polynomal specal type of topologcal space assocated wth these MOD semgroups. The reader s expect to fnd MOD natural neutrosophc fnte complex number subset polynomal specal topologcal space zero dvsors, nlpotents and dempotents. The reader s left wth the task of fndng order of B. Next we proceed onto gve several problem for the nterested reader.

72 MOD Subset Semgroups Under 71 PROBLEMS 1. Study the MOD subset semgroup S ={P(Z n ) under }. ) Can S have MOD subset deals? ) ) v) Can S have MOD subset subsemgroups? Can we say f n s a composte number S wll have more deals? s () true? Justfy your result. Can we say f n s a prme S wll have less number of deals? s (v) true? Justfy your clam. v) Prove every subset s a MOD subset semgroup and s not an deal.. Let B = {P(Z 8 ), } be the MOD subset semgroup. Study questons () to (v) of problem (1) for ths B.. Let D = {P(Z 5 ), } be the MOD subset semgroup. Study questons () to (v) of problem (1) for ths D.. Can there be any structure dfference between B and D of problems and respectvely (as the operaton s only }? 5. Let M = {P(Z 18 ), } be the MOD subset semgroup. Study questons () to (v) of problem (1) for ths M.. Can there be structural dfference between B n problem () and ths M n problem (5)? 7. Can there be any structural dfference between D n problem () and M s problem (5)?

73 7 MOD Natural Neutrosophc Subset Topologcal 8. Can we say what ever be n ( n < ) n P(Z n ) under the bnary operaton they behave unformly mmateral of n odd or n even or n prme or n = p t ; t and p a prme? Prove your clam. 9. Let W = {P(Z n ), } be the MOD subset semgroup under the operaton. Study questons () to (v) of problem (1) for ths W. 1. Let V ={P(Z 1 ), } be the MOD subset semgroup under the operaton. Study questons () to (v) of problem (1) for ths V. 11. Let R = {P(Z 7 ), } be the MOD subset semgroup. Study questons () to (v) problem (1) for ths R. 1. Let E = {P(Z 8 ), } be the MOD subset semgroup under. Study questons () to (v) of problem (1) for ths E. 1. Compare R of problem (11) wth V of problem (1). 1. Fnd the dfference f t exst n structure between R of problem (11) and E of problem (1). 15. Compare the MOD subset semgroups n problems (1) and (1). 1. Let T ={P(Z 15 ), +} be the MOD subset semgroup. ) Show T has only few MOD subset subsemgroups. ) ) s T a S-semgroup? Can T have S-subsemgroups?

74 MOD Subset Semgroups Under 7 v) Can T have S-deals? v) Obtan any other specal feature assocated wth T. 17. Compare the MOD subset semgroups {P(Z( n ), }, {P(Z n ), } and {P(Z n ), +} among themselves. Prove {P(Z n ), +} has the mnmum number of MOD subset subsemgroups n comparson wth {P(Z n ), } and {P(Z n ), }. 18. Let N = {P(Z 79 ), +} be the MOD subset semgroup. ) Study questons () to (v) of problem (1) for ths N. ) Prove N has less number of MOD subset subsemgroups n comparson wth T. 19. Let U = {P(Z ), +} be the MOD subset semgroup. ) Study questons () to (v) of problem (1) for ths U. ) Compare Uwth N of problem (18). ) Compare U wth T of problem (1).. Let Y = {P(Z 8 ), } be the MOD subset semgroup under product; ) s Y a S-subset semgroup? ) ) v) Can Y have MOD subset zero dvsors? Can Y have MOD subset S-zero dvsors? Can Y have MOD subset dempotents? v) Can Y have MOD subset S-dempotents? v) v) Does Y contan MOD subset nlpotents? Fnd all MOD subset subsemgroups of Y whch are not deals of Y.

75 7 MOD Natural Neutrosophc Subset Topologcal v) Fnd all MOD subset deals of Y. x) Can Y have MOD subset S-deals? x) Obtan any other specal feature assocated wth Y. 1. Let C = {P(Z ), } be the MOD subset semgroup. ) Study questons () to (x) of problem () for ths C. ) Prove C s a S-subset semgroup. ) Prove C has less number of MOD subset subsemgroups and deals n comparson wth Y of problem (). v) Prove C has no nontrval MOD subset zero dvsors, dempotents and nlpotents. v) Obtan any other specal feature assocated wth C.. Let G = {P( Z 8 5 ), } be the MOD subset semgroup. ) Study questons () to (x) of problem for ths G. ) Prove G has more MOD subset nlpotents than Y n problem and C n problem 1. ) Prove G has more deals than C n problem 1. v) Compare Y of problem wth ths G. v) Compare C of problem 1 wth ths G.. Let P = {P(Z n ); n < } be the power set of Z n. {P,}, {P, }, {P, +} and {P, } are the four dstnct MOD subset semgroups of P.

76 MOD Subset Semgroups Under 75 Let P o = {P,, }, P = {P, +, }, P = {P, +, }, P = {P, +, }, P = {P,, } and P = {P,, } be the sx MOD subset specal type of topologcal spaces. ) Whch MOD subset specal type of topologcal space has maxmum number of MOD subset specal type of topologcal subspaces? ) ) v) Whch of them gven by () are MOD subset deals? Fnd all MOD subset specal type of topologcal zero dvsors, nlpotents and dempotents of P o, P, P, P, P and P. Can P have MOD subset deals?. Let V = {P(Z )} be the MOD subset collecton. V o, V, V, V, V and V be the MOD subset specal type of topologcal spaces. ) Study questons () to (v) of problem () for ths V. ) Obtan any other specal feature assocated wth V. 5. Let W = {P(Z 5 )} be the MOD subset collecton. Let W o, W, W, W, W and W be the MOD subset specal type of topologcal spaces assocated wth W. ) Study questons () to (v) of problem () for ths W. ) Compare W wth V of problem (). ) Obtan all specal features assocated wth W.. Let R = {P(Z )} be the MOD subset; R o, R, R, R, R and R be the MOD subset specal type of topologcal spaces assocated wth R.

77 7 MOD Natural Neutrosophc Subset Topologcal ) Compare R wth W and V of problem (5) and () respectvely. ) Study questons () to (v) of problem () for ths R. 7. Let S 1 = {P(Z 1 )} be the MOD neutrosophc subset collecton. {S 1, } be the MOD neutrosophc subset semgroup. ) Study questons () to (v) of problem (1) for ths S 1. ) Fnd all specal features assocated wth ths S 1. ) How s ths S 1 dfferent from S n problem 1?. 8. Let R 1 = {P(Z 19 ), } be the MOD neutrosophc subset semgroup. ) Study questons () to (v) of problem (1) for ths R 1. ) Compare R 1 wth S 1 of problem (7). 9. Let D 1 = {P(Z ), } be the MOD neutrosophc subset semgroup. ) Study questons () to (v) problem (1) for ths D 1. ) Compare D 1 wth R 1 of problem 8. ) Compare D 1 wth S 1 of problem 7.. Let L 1 = {P(Z 1 ), } be the MOD neutrosophc subset semgroup. Study questons () to (v) of problem (1) for ths L Let X 1 = {P(Z 9 ), } be the MOD neutrosophc subset semgroup.

78 MOD Subset Semgroups Under 77 ) Study questons () to (v) of problem 1 for ths X 1 ) Compare ths X 1 wth L 1 of problem.. Let Y 1 = {P(Z 1 ), } be the MOD neutrosophc subset semgroup. ) Study questons () to (v) of problem (1) for ths Y 1. ) Compare ths Y 1 wth X 1 of problem 1. ) Compare ths Y 1 wth L 1 of problem.. Let E 1 = {P(Z ), } be the MOD subset neutrosophc semgroup under +. ) Study questons () to (v) of problem (1) for ths E 1.. Compare G = {P(Z n ), } wth E = {P(Z n ), } and F = {P(Z n ), }; n <. 5. Let M = {P(Z 7 ), } be the MOD subset neutrosophc semgroup. Study questons () to (v) of problem 1 for ths M.. Let P = {P( Z 7 subset semgroup. ), } be the MOD neutrosophc ) Study questons () to (v) of problem(1) for ths P. ) Compare ths P wth M of problem 5. ) Compare ths P wth problem E 1 of. v) Compare problem E 1 of wth problem M of 5.

79 78 MOD Natural Neutrosophc Subset Topologcal 7. Let W = {P(Z 8 ), } be the MOD neutrosophc subset semgroup under. ) Study questons () to (x) of problem () for ths W. 8. Let M 5 = {P(Z 17 ), } be the MOD neutrosophc subset semgroup. ) Study questons () to (x) of problem () for ths M 5. ) Compare ths M 5 wth W of problem Let N 8 = {P( Z 5 subset semgroup. ), } be the MOD neutrosophc ) Study questons () to (x) of problem () for ths N 8. ) Compare ths N 8 wth M 5 of problem 8. ) Compare ths N 8 wth W of problem 7.. Let T = {P(Z n ); n < } be the power set of Z n. Let {T, +}, {T, }, {T, } and {T, } be MOD neutrosophc subset semgroups. T o = {T,, }, T = {T, +, }, T = {T, +, }, T = {T, +, }, T = {T,, } and T = {T,, } be the MOD neutrosophc subset specal type of topologcal spaces. Study questons () to (v) of problem() for ths T. 1. Let W = {P(Z 9 )} be the MOD neutrosophc subsets of Z 9, W o, W, W, W, W and W be the MOD neutrosophc subset specal type of topologcal spaces assocated wth the power set P(W). Study questons () to (v) of problem () for ths W.

80 MOD Subset Semgroups Under 79. Let J = {P(Z 1 )} be the MOD neutrosophc subsets of Z 1. J o, J, J, J, J and J be the MOD neutrosophc subset specal type of topologcal spacesof the power set J. ) Study questons () to (v) of problem () for ths J. ) Compare ths J wth W of problem.. Let K = {P(Z 9 )} be the MOD neutrosophc subset specal type of topologcal spaces constructed usng K. ) Study questons () to (v) of problem () for ths K. ) Compare K wth J n problem (). ) Compare K wth W n problem (1). v) Compare J n problem () and W n (1).. Let H = {P(Z )} be the MOD fnte complex number subsets collecton. H o, H, H, H, H and H be the MOD fnte complex number subset specal topologcal spaces. ) Study questons () to (v) of problem () for ths H. ) Fnd all specal features enjoyed by H. ) Compare H wth K of problem () v) Compare H wth J of problem (). 5. Let B = {P(C(Z ))} be the MOD fnte complex number subsets collecton. B o, B, B, B, B and B be the MOD fnte complex number subset specal type of topologcal spaces usng B. ) Study questons () to (v) of problem () for ths B.

81 8 MOD Natural Neutrosophc Subset Topologcal ) Compare ths B wth H of problem (). ) Compare ths B wth K of.. Let D = {P(C(Z 18 ))} be the MOD fnte complex number subset collecton. D o, D, D, D, D and D be the MOD fnte complex number subset specal topologcal spaces. ) Study questons () to (v) of problem (). ) Compare D wth B of problem (5) ) Compare D wth H of problem (). 7. Work wth all propertes of MOD subset fnte complex number semgroups bult usng P 1 = {P(C(Z ))}, P = {P(C(Z ))}, P = {P(C( Z 7 ))} under the,, + and. 8. Let E = {P(Z n g)} be the MOD subset dual number collecton from Z n g = {a + bg / a, b Z n, g = }. {E, } be the MOD subset dual number semgroup. Study questons () to (v) of problem (1) for ths E. 9. Let V = {P(Z 1 g), } be the MOD subset dual number semgroup. ) Study questons () to (v) of problem (1) for ths V. ) Compare ths V wth E of problem Let W = {P (Z g), } be the MOD subset dual number semgroup. Study questons () to (v) of problem 1 for ths E. 51. Let P = {P ( Z g), } be the MOD subset dual 7 number semgroup.

82 MOD Subset Semgroups Under 81 ) Study questons () to (v) of problem 1 for ths P. ) Compare ths P wth V of problem 9 ) Compare ths P wth E of problem Let Z = {P (Z n g), } be the MOD subset dual number semgroup. ) Study questons () to (v) of problem (1) for ths Z. ) Compare Z wth the MOD subset dual number semgroup n whch s replaced by. ) Obtan all specal features enjoyed by Z. 5. Let A = {P (Z g), } be the MOD subset dual number semgroup. ) Study questons () to (v) of problem (1) for ths A. ) Compare A wth the MOD semgroup M = {P (Z g), }. 5. Let B = {P (Z 5 g), } be the MOD subset dual number semgroup ) Study questons () to (v) of problem (1) ths B. ) Compare B wth A of problem Let C = {P (Z 11 g), } be the MOD subset dual number semgroup. ) Study questons () to (v) of problem (1) for ths C. ) Compare ths C wth A of problem 5. ) Compare ths C wth B of problem 5.

83 8 MOD Natural Neutrosophc Subset Topologcal 5. Let D = {P (Z n g), } be the MOD dual number subset semgroup. ) Study propertes () to (x) of problem for ths D. ) Compare the D wth D 1 = {P (Z n g), }, D = {P (Z n g), } and D = {P (Z n g), }. 57. Let E = {P (Z g), } be the MOD subset dual number semgroup. ) Study propertes () to (x) of problem () for ths E. ) ) Prove E has more numbers of zero dvsors and nlpotent subsets than Y n problem. Obtan any other specal feature assocated wth E. 58. Let F = {P (Z 7 g), } be the MOD dual number subset semgroup. ) Study questons () to (x) of problem () for ths F. ) Compare ths F wth E of problem 57. ) Prove E n problem 57 has more zero dvsor than F. 59. Let G = {P (Z 19 g), } be the MOD subset dual number semgroup. ) Study questons () to (x) of problem () for ths G. ) Compare ths G wth (F) of problem 58. ) Compare ths G wth (E) of problem 57. v) Compare ths G wth (D) of problem 5.

84 MOD Subset Semgroups Under 8 v) Prove G has more number of nlpotents than D, E and F of problems 5, 57 and 58 respectvely.. Let H = {P (Z n g)} be the MOD dual number subset collecton H o, H, H, H, H and H be the MOD subset dual number specal type of topologcal spaces. ) Prove all the sx spaces are dstnct. ) ) v) Prove H has more number of MOD dual number subset topologcal nlpotents where every be n, n <. Prove H has more number of MOD dual number subset topologcal zero dvsors what be n; n <. Study questons () to (v) of problem () for ths H. 1. Let = {P (Z g)} be the MOD dual number subset collecton o,,,, and be the MOD dual number specal type of topologcal spaces. ) Study questons () to (v) of problem () for ths H. ) Prove H has more number of dempotents.. Let J = {P (Z 7 g)} be the MOD dual number collecton J o, J, J, J, J and J be the MOD subset dual number specal type of topologcal spaces. ) Study questons () to (v) of problem () for ths J. ) Prove J has less number of MOD subset dual number topologcal dempotents, zero dvsors and nlpotents than n problem 1.

85 8 MOD Natural Neutrosophc Subset Topologcal. Let K = {P (Z 1 g)} be the MOD subset dual number assocated topologcal spaces K o, K, K, K, K and K. ) Study questons () to (v) of problem () for ths K. ) Prove K has more number of MOD subset dual number topologcal nlpotents. ) Compare K wth J of problem. v) Compare K wth of problem 1.. Let L = {P (Z n h)} be the MOD specal dual lke number subsets from Z n h = {a + bh / a, b Z n, h = h}; n <. ) Prove {L, }, {L, }, {L, } and {L, +} are four dstnct MOD specal dual lke number subset semgroups. ) Prove L = {L,+, }, L o = {L,, }, L = ) v) {L, +, L = {L, +, }, L = {L,, } and L = {L,, } are sx dstnct MOD specal dual lke number subset specal type of topologcal spaces. Study for ths {L, } and {L, } questons () to (v) of problem (1). Study for {L, +} questons () to (v) of problem 1. v) Study questons () to (x) of problem () for ths {L, }. v) Fnd all MOD subset specal dual lke number specal topologcal subspace zero dvsors, dempotents and nlpotent subsets.

86 MOD Subset Semgroups Under Let M = {P (Z h)} be the MOD specal dual lke number subsets M o, M, M, M, M and M be the MOD specal dual lke number subsets specal types of topologcal spaces assocated wth the MOD specal dual lke number subset semgroups {M, +}, {M, }, {M, } and {M, }. ) Study questons () to (v) of problem for ths M. ) Derve all specal features assocated wth M.. Let N = {P (Z 1 h)} be the MOD specal dual lke number subsets N o, N, N, N, N and N be the MOD specal dual lke number subset specal type of topologcal spaces assocated wth {N, }, {N, +}, {N, } and (N, } ) Study questons () to (v) of problem for ths N. ) Compare ths N wth M of problem (5). ) Prove N has less number MOD topologcal dempotents zero dvsors and nlpotents n comparson wth M problem Let O = {P ( Z 5 h)} be the MOD specal dual lke 5 number subsets. Let O o, O, O, O, O and O be the MOD specal dual lke number subset specal type of topologcal spaces assocated wth the MOD specal dual lke number subset semgroups {O,+}, {O, }, {O, } and {O, }. ) Study questons () to (v) of problem () for ths O. ) Compare ths O wth N of problem ().

87 8 MOD Natural Neutrosophc Subset Topologcal ) Compare ths O wth M of problem (5). v) Prove O has more number of MOD topologcal nlpotents than M and N n problems () and () respectvely. 8. Let P = {P (Z n k)} be the MOD specal quas dual number subsets from Z n k = {a + bk / a, b Z n, k = (n 1)k}. {P, +}, {P, }, {P, } and {P, } be the MOD specal quas dual number subset semgroups. Let P o, P, P, P, P and P be the MOD specal quas dual number subset specal type of topologcal spaces. ) Study questons () to (v) of problem for ths P. ) ) Study questons to (v) of problem () for ths P. Obtan any other specal feature assocated wth P. 9. Let Q = {P (Z 1 k)} be the MOD specal quas dual number subset. {Q, +}, {Q, }, {Q, } and {Q, } be the MOD specal quas dual number subset semgroups. Let Q o, Q, Q, Q, Q and Q be the MOD specal quas dual number subset specal type of topologcal space bult usng the MOD subset semgroups. Study questons () to () of problem (8) for ths Q. 7. Obtan all specal features assocated wth each of these P(Z n g ) the collecton of all subsets of MOD natural neutrosophc dual numbers. 71. Let R = {P (Z 5 k)} be the MOD specal quas dual number subsets of Z 5 k. {R, +}, {R, }, {R, } and {R, } be the MOD specal quas dual number subset semgroups. R o, R, R, R, R and R be the

88 MOD Subset Semgroups Under 87 MOD specal quas dual number subset specal type of topologcal spaces assocated wth the four semgroups. ) Study questons () to () of problem (8) for ths R. ) Obtan any other specal feature assocated wth R. ) Compare R wth problem Q of problem (9) 7. Let S = {(a 1, a, a, a, a 5 ) / a P(Z 1 ); 1 5} be the MOD subset matrx collecton. {S, }, {S, }, {S, +} and {S, } be the MOD subset matrx semgroups. S o = {S,, }, S = {S, +, }, S = {S, +, }, S = {S, +, }, S = {S,, } and S = {S,, } be the MOD subset matrx specal type of topologcal spaces assocated wth the MOD subset semgroups. ) Can {S, }, {S, }, {S, } and {S, +} have deals? ) ) v) Fnd all MOD subset matrx subsemgroups whch are not deals. Whch of the four MOD subset semgroups are S-semgroup? What are the MOD subset topologcal zero dvsors n S, S and S? v) Fnd all MOD subset topologcal dempotents n S. v) Can the MOD subset topologcal spaces have S- zero dvsors? Justfy! v) v) Can these MOD subset topologcal spaces have S-dempotents? Can the MOD subset specal type of matrx topologcal spaces contan topologcal nlpotent elements?

89 88 MOD Natural Neutrosophc Subset Topologcal x) Prove S cannot have MOD matrx subset-strong topologcal deals. x) Prove S can have MOD matrx subset strong specal type of topologcal subspaces. a1 a a 7. Let W = { / a P(Z 1 ), 1 } be the matrx a a 5 a subsets collecton {W, }, {W, }, {W, n } and {W, +} be the MOD matrx subset semgroups. S o, S, S, S n, S n and be the MOD matrx subset specal type of topologcal spaces. ) Study questons () to (x) of problem (7) for ths S. ) Compare S n (7) wth ths W. a a a a a a a a 7. Let P = { a a a a a a a a n S / a P(Z 8 ), 1 } be the MOD subset matrx collecton {P, n }, {P, }, {P, } and {P, } and {P, } be the MOD matrx subset semgroups. P o, P, P, P, P, P, n P, n P and P n be the MOD matrx subset specal type of topologcal spaces assocated wth these fve semgroups. ) Fnd deals n these fve MOD semgroups. ) Show {P, } and {P, n } have zero dvsors and dempotents.

90 MOD Subset Semgroups Under 89 ) v) Can {P, } and {P, n } have nlpotents? Can {P, } and {P, n } have S-zero dvsors? v) Can {P, } and {P, n } be S-semgroups? v) v) v) x) Prove {P, +} s always a S-semgroup. Prove the MOD subset matrx specal type of topologcal spaces has no strong deals. Prove MOD subset matrx specal type of topologcal spaces have strong topologcal subspaces and none of them are topologocal deals. Prove MOD subset topologcal matrx spaces under usual product has left zero dvsors whch are not rght zero dvsors. x) Prove MOD subset topologcal matrx spaces under has left deals whch are not rght deals and vce versa. x) x) Can MOD subset topologcal matrx spaces have a) S-zero dvsors under or o? b) S-dempotents under or o? c) S-deals under or n? d) S-topologcal subset matrx subspaces whch are not S-deals? (By MOD S-topologcal matrx subspaces we mean under both the operatons of the MOD specal type of subset spaces the assocated subsemgroups must be S-subsemgroups). Obtan any other specal features enjoyed by these 1 dstnct MOD specal type of subset matrx topologcal spaces.

91 9 MOD Natural Neutrosophc Subset Topologcal a1 a a 75. Let S 1 = { / a P(C(Z 11 )), 1 } be the MOD a a 5 a matrx subset fnte complex number collecton {S 1, n }, {S 1, }, {S 1, } and {S 1, } are the four MOD subset matrx semgroups and related wth these MOD subset matrx semgroups we have MOD subset matrx specal topologcal spaces. ) Study questons () to (x) of problem (7) for ths S 1 ) Compare ths S 1 wth W of problem (7) ) What s the role played by the fnte complex number subset collecton? a1 a a 7. Let M = { a a5 a / a P(C(Z )); 1 9} be a7 a8 a 9 the MOD subset matrx fnte complex number collecton ) Study questons () to (x) of problem (7) for ths M. ) Dstngush ths M from P of problem 7. a1 a a a 77. Let T = { a5 a a 7 a / a P(C(Z 1 )); 1 8} 8 be the MOD subset matrx fnte complex number collecton. ) Study questons () to (x) of problem (71) for ths T.

92 MOD Subset Semgroups Under 91 ) Dstngush between S of problem 7 and that of T. a a a a a a 78. Let N = { a a a a a a / a P(Z 1 1 1} be the collecton of MOD subset matrx neutrosophc collecton. ) Study questons () to (x) of problem (7) for ths N. a a 79. Let O = { a a a a 1 a 7 a 8 a 9 a 5 1 / a P(Z 19 ), 1 1} be the collecton of all MOD subset matrx. ) Study questons () to (x) of problem 7 for ths O. ) Compare N and S problems 78 and 75 respectvely wth O. a a a a a a a a a a 8. Let B = { a a a a a a a a a a a a a a a / a P(Z 8 ); 1 5} be the MOD subset matrx neutrosophc collecton.

93 9 MOD Natural Neutrosophc Subset Topologcal ) Study questons (x) to (x) of problem 7 for ths B. ) Compare B wth problem 7 of M. a1 a a 81. Let P = { / a P(Z 19 h), 1 } be the a a 5 a MOD subset matrx collecton dual lke number collecton. ) Study questons () to (x) of problem (7) for ths A. ) Compare ths A wth O of problem (79). a a a a a a a a 8. Let D = { a a a a a a a a / a P(Z h), 1 1} be the MOD subset matrx specal dual lke number collecton. ) Study questons () to (x) problem (7) for ths D. ) Obtan all specal features assocated wth D.

94 MOD Subset Semgroups Under 9 a a a 8. Let E = { a a a a 1 a a 5 a 7 8 a 9 1 a 11 1 / a P(Z 5 g); 1 1} be the MOD subset matrx dual number collecton ) Study questons () to (x) of problem (7) for ths E. ) Prove E has more number of zero dvsors and nlpotents. ) Derve all specal features assocated wth E. a a a a a a a a a a 8. Let F = { a a a a a a a a a a a a a a a / a P(Z 1 g), 1 5} be the MOD square matrx dual number subset collecton. ) Study questons () to (x) of problem (7) for ths F. ) Compare ths F wth D n problem 8. a a a a a 85. Let G = { a a a a a / a P(Z 8 k); 1 1} be the MOD subset matrx of specal quas dual number collecton.

95 9 MOD Natural Neutrosophc Subset Topologcal ) Study questons () to (x) of problem (7) for ths G. ) Enlst all the specal features enjoyed by ths G. a a a a a a a a a a a a a a a a a a 8. Let H = { a a a a a a a a a a a a a a a a a a /a P(Z k), 1 } be the MOD specal quas dual number matrx subset collecton. ) Study questons () to (x) of problem 7 for ths H. ) Compare ths H wth F of problem Let S = {P( Z 1 ) / Z 1 = {a + 1 t where a Z 1, t Z 1 s a zero or a zero dvsor or a nlpotent or an dempotent} be the MOD natural neutrosophc subset collecton. {S, +}, {S, }, {S, and {S, } be the MOD natural neutrosophc subset semgroup. S o, S, S, S, S and S be the MOD natural neutrosophc subset specal type of topologcal spaces. ) Study questons () to (v) of problem (1) for ths S. ) Dstngush S from R ={P(Z 1 )} where Z 1 s replaced by Z. 1

96 MOD Subset Semgroups Under 95 ) Does the collecton of MOD natural neutrosophc elements of Z1 enjoy any other specal property? v) What can be the probable applcatons of MOD natural neutrosophc specal type of topologcal spaces? v) Are these MOD natural neutrosophc specal type of topologcal spaces dscrete? Justfy your clam. v) Show the MOD natural neutrosophc specal type of topologcal subset spaces has strong subspaces but has no strong deals. v) Show MOD natural neutrosophc specal type of topologcal subset zero dvsors, dempotents and nlpotent. v) Obtan all other specal features assocated wth these MOD natural neutrosophc subset specal type of topologcal spaces. 88. Let S 1 = {P( Z )} be the collecton of MOD natural neutrosophc subsets collecton. ) Study questons () to (v) of problem (87) for ths S 1 ) Compare S of problem 87 wth ths S Let B = {P(Z 18 = {a + b + t / a, b Z 18, t Z 18 where t s an ndetermnate or an dempotent or nlpotent or a zero dvsor, = } be the collecton of all MOD natural neutrosophc-neutrosophc subsets. {B, +}, {B, }, {B, } and {B, } be the MOD natural neutrosophc semgroups. Let B o, B, B, B, B and B be the MOD natural neutrosophc-neutrosophc specal type of subset topologcal spaces bult usng {B, +}, {B, }, {B, } and {B, }.

97 9 MOD Natural Neutrosophc Subset Topologcal ) Study questons () to (v) of problem (87) for ths B. ) Compare B wth S of problem Let D = {P(Z 7 )} be the collecton of MOD natural neutrosophc-neutrosophc subset specal type of topologcal spaces, D o, D, D, D, D and D related wth the MOD natural neutrosophc-neutrosophc subset semgroups, {D, }, {D, }, {D, } and {D, }. ) Study questons () to (v) of problem (87) for ths D. ) Compare S 1 of problem 88 wth D. 91. Let E = {P(C (Z 8 ) = {collecton of all subsets from C c (Z 8 ) = {a + b F + t / a, b Z 8, ; t C(Z F 8), s a zero dvsor or nlpotent or an dempotent} be collecton of all MOD natural neutrosophc fnte complex number subets. E o, E, E, E, E and E be the MOD natural neutrosophc fnte complex number subset specal type of topologcal spaces. ) Study questons () to (v) of problem (87) for ths E. ) Compare ths E wth D of problem Let F = {P(C (Z 1 ))} be the MOD natural neutrosophc fnte complex number subsets F o, F, F, F, F and F be the MOD natural neutrosophc fnte complex number subset specal type of topologcal spaces. ) Study questons () to (v) of problem 87 for ths F. ) Compare ths F wth E of problem 91.

98 MOD Subset Semgroups Under Let H = {P(Z g ) = {collecton of all subsets from MOD natural neutrosophc dual number Z g = {a + g bg + t / a, b Z, g =, t Z g; where t s a nlpotent or dempotent or a zero dvsor} be the MOD natural neutrosophc fnte complex number subsets. H o, H, H, H, H and H be the MOD natural neutrosophc fnte complex number subset specal topologcal spaces assocated wth H. ) Study questons () to (v) of problem 87 for ths H. ) Compare ths H wth E of problem 91. ) Gve all the specal features assocated wth H. 9. Let = {P(Z 7 g )} be the MOD natural neutrosophc dual number subsets collectons, o,,,, and be the MOD natural neutrosophc dual number subset specal type of topologcal spaces. ) Study questons () to (v) of problem 87 for ths. ) Compare ths wth H of problem Let J = {P(Z 1 h )} = {collecton of all MOD natural neutrsophc specal dual lke number subsets Z 1 h h = {a + bh + t / a, b Z 1, h = h, t Z 1 h s a zero dvsor or nlpotent or an dempotent of Z 1 h} be the MOD natural neutrosophc specal dual lke number subsets collecton. J o, J, J, J, J and J be the MOD natural neutrosophc specal dual lke number subset specal topologcal spaces. ) Study questons () to (v) of problem (87) for ths J. ) Compare of problem 9 wth ths J.

99 98 MOD Natural Neutrosophc Subset Topologcal ) Obtan all specal features assocated wth J. 9. Let K = {P(Z 8 k )} = {collecton of all subsets from Z 8 k, = {a + bk + / a, b Z 8, k = 7k, t Z 8 h where t s an dempotent or nlpotent or a zero dvsor}} be the MOD natural neutrosophc specal quas dual number subset collecton. K o, K, K, K, K and K be the MOD natural neutrosophc specal quas dual number subset specal type of topologcal spaces. ) Study questons () to (v) of problem (87) for ths K. ) Compare J of problem 95 wth ths K. ) k t Obtan all specal features assocated wth ths K. 97. Let L = {P (Z k )} = {collecton of all natural neutrosophc specal quas dual number subsets from {Z k = {a + bk + / a, b Z, K = k, t Z k s a zero dvsor or dempotents or nlpotents} MOD natural neutrosophc specal quas dual number subset collectons. L o, L, L, L, L and L be the MOD natural neutrosophc specal quas dual number subset specal type of topologcal spaces. ) Study questons () to (v) of problem (87) for ths L. ) k t Obtan all specal features assocated wth ths L. 98. Let N = {P(M)} = {collecton of all matrx subsets of

100 MOD Subset Semgroups Under 99 a1 a M wth entres from Z 1, where M = a / a Z 1 }; 1 a a 5 5}} be the MOD subset matrx collecton. ) Study questons () to (v) of problem 89 where {N, +}, {N, n }, {N, } and {N, } are the assocated MOD matrx subset semgroup. ) Prove these related topologcal has MOD subset matrx topologcal nlpotents, MOD subset matrx topologcal zero dvsors and dempotents. ) Derve all specal features assocated wth N. 99. Let P = {P(R)} = {collecton of all subset matrces from matrces from a1 a a a R = { a5 a a7 a8 / a C (Z 1 ); 1 8}} be the MOD natural neutrosophc fnte complex number subset matrx collecton. R o, R, R, R, n R and n R be the MOD natural neutrosophc fnte complex number subset matrx specal type of topologcal spaces assocated wth the MOD natural neutrosophc fnte complex number subset matrx semgroups {R, n }, {R, }, {R, and {R, +}. ) Study questons () to (v) of problem (89) of R. n

101 1 MOD Natural Neutrosophc Subset Topologcal ) Study the specal feature enjoyed by R o, R n and so on. ) Compare N of problem 98 wth ths T. 1. Let T = {P(W)} = {Collecton of all subsets from a a a a a a a a W = { a a a a a a a a / a Z 11 g ; 1 1}} be the MOD natural neutrosophc dual number subset matrx collectons, T o, T, T, T, T, T, T, n T, T n and T n be the MOD natural neutrosophc dual number matrx subset specal type of topologcal spaces assocated wth {T, }, {T, }, {T, +}, {T, n } and {T, }. ) Study questons () to (v) of problem (89) for ths T. ) Compare ths T wth R of problem (99). 11. Let S = {P(V)} = {collecton of all subset matrces from n a a a a a a V ={ a a a a a a / a Z 8 k, 1 1}} be the MOD natural neutrosophc specal quas dual number subset matrx collecton.

102 MOD Subset Semgroups Under 11 ) Study questons () to (v) of problem (89) for ths V. ) Compare ths V wth T of problem (1). 1. Let a a a a a B = { a a a a a / a P(Z 1 ); 1 1} be the MOD neutrosophc matrces wth entres from subsets of Z 1 ; collecton. {B, +}, {B, }, {B, } and {B, n } be the MOD neutrosohc subset entres matrx subsemgroups. B o, B, B, B, B and B be the dstnct MOD neutrosophc subset specal type of topologcal spaces. ) Study questons () to (v) of problem (89) for ths B. 1. Let D = { a1 a a a a 5 a a 7 / a P(Z 1 g ); 1 7} be the MOD natural neutrosophc dual number matrces wth subset entres. ) Study questons () to (v) of problem (89) for ths D.

103 1 MOD Natural Neutrosophc Subset Topologcal ) Compare ths D wth B of problem (1). ) 15. Let Enumerate all the specal features assocated wth ths D. 1. Let a a E = { a a a a 1 a a 5 a 7 8 a 9 1 / a P(Z 8 k ); 1 1} be the MOD natural neutrosophc specal quas dual number subset matrx collecton. E o, E, E, E, E and E be the MOD natural neutrosophc specal quas dual number subset matrx specal type of topologcal space assocated wth {E, +}, {E, }, {E, } and {E, }. ) Study questons () to (v) of problem (89) for ths E. ) Compare ths E wth D of problem (1). M = { a a a a a a a a a a a a a a a a a a a a a a a a a / a P(C Z )) = {collecton of all subsets from C (Z ) = {a + b F + / a, b Z, = 9, t C(Z ) where t s a zero c t F dvsor or dempotent or nlpotent of C(Z ); 1 5} be the MOD natural neutrosophc fnte complex number

104 MOD Subset Semgroups Under 1 subset matrx collecton. M o, M, M, M, M, M, M n, M, n n M and n M be the MOD natural neutrosophc fnte complex number subset matrx specal type of topologcal spaces assocated wth the MOD natural neutrosophc fnte complex number subset matrx semgroup (M, +},{M, }, {M, }, {M, n } and {M, }. ) Prove M, M, M and M are MOD natural n neutrosophc fnte complex number subset matrx specal type of toplogcal spaces whch are non commutatve. ) ) Show these four spaces M, M, M and M have (a) Rght MOD subset matrx n topologcal zero dvsors whch are left zero dvsors and vce versa. Fnd all MOD subset rght deals of these four spaces whch are not left deals and vce versa. 1. Let G = { ax / a P( Z 8 )} = {collecton of polynomals wth coeffcents from the subsets of Z 8 = {a + t / a Z 8 and t Z 8 s such that t s a nlpotent or an dempotent or a zero dvsor of Z 8 }} be the MOD natural neutrosophc subset coeffcent polynomals. {G, +}, {G, }, {G, } and {G, } are MOD natural neutrosophc subset polynomal semgroups. G o, G, G, G, G and G are the MOD natural neutrosophc subset coeffcent polynomal specal type of topologcal spaces assocated wth semgroups.

105 1 MOD Natural Neutrosophc Subset Topologcal ) Fnd all MOD zero dvsor subset polynomals of the topologcal spaces G, G and G. (a) f x G s a MOD topologcal zero dvsor then the same x n G and G are also MOD subset topologcal zero dvsor. ) ) v) Fnd a strong MOD neutrosophc subset coeffcent polynomal specal type of topologcal subspace. Prove there does not exst MOD neutrosophc subset coeffcent polynomal specal type of topologcal deals. Show f W G o s a MOD subset coeffcent polynomal subspace of G o then W G s not 17. Let a subspace G or G or G and G or G v) Obtan any other specal feature assocated wth these spaces. H = { a x / a P(Z 1 g)} be the MOD dual number subset coeffcent polynomal collecton.{h, +}, {H, }, {H, } and {H, } be the MOD dual number subset coeffcent polynomal semgroups. H o, H, H, H, H and H be the related MOD dual number subset coeffcent polynomal specal type of topologcal spaces assocated wth these MOD semgroups. ) Study questons () to (v) of problem (1) for ths H.

106 MOD Subset Semgroups Under Let ) Compare ths H wth G of problem 15. = { ax / a P(C (Z 7 ))} be the MOD natural neutrosophc fnte complex number subset coeffcent polynomals. o,,,, and be the MOD natural neutrosophc fnte complex number subset coeffcent polynomal specal type of topologcal spaces assocated wth. ) Study questons () to (v) of problem (1) for ths. ) Compare wth ths the H of problem 1. ) Compare wth ths G of problem Let J = P(S[x]) = {collecton of all subsets from S[x] = { a x / a Z 18 )} be the MOD neutrosophc subsets of polynomal. Let {J, }, {J, +}, {J, } and {J, } be the MOD neutrosophc subset polynomal semgroups. Let J o, J, J, J, J and J be the MOD natural neutrosophc subsets n polynomal specal type of topologcal spaces. ) Fnd all strong MOD subset specal type of topologcal spaces of all the sx spaces. ) Prove n general f Y s a MOD subspace of any one of the topologcal spaces then n general need not be the MOD subspaces of other MOD topologcal spaces.

107 1 MOD Natural Neutrosophc Subset Topologcal ) v) Obtan MOD nlpotents and zero dvsors of these topologcal spaces. Enumerate any specal property assocated wth J. v) Can J have MOD dempotents, justfy your clam? 11. Let L = {P(M[x])} = {collecton of all polynomal subsets from M[x] = { ax / a Z 8 k = Pa + bk / a, b Z 8, k 111. Let = 7k}}} be the MOD specal quas dual number polynomal subsets from P(M[x]). ) Study questons () to (v) of problem (18) for ths L ) Compare ths L wth J n problem (18). 9 P = { ax / a P(Z 8 ), x 1 = 1} be the MOD neutrosophc subset coeffcent fnte degree polynomal {P, +}, {P, +}, {P, } and {P, } be the MOD neutrosophc coeffcent fnte degree polynomal semgroups. P o, P, P, P, P and P be the MOD neutrosophc coeffcent polynomal specal type of topologcal spaces assocated wth the four semgroups. ) Prove all these topologcal spaces are of fnte order. ) Fnd all MOD subset polynomal coeffcent strong spaces.

108 MOD Subset Semgroups Under 17 ) v) Can these MOD topologcal spaces assocated wth P have strong deals? Justfy. Fnd all MOD topologcal zero dvsors, nlpotents and demoptents (f any) n P, P, and P Let R = { ax / a P(Z 1 g) (or C (Z 1 )); x 19 = 1} be the MOD natural neutrosophc dual number (fnte complex number) subset coeffcent polynomals of fnte order. R o, R, R, R, R and R be the MOD subset natural neutrosophc polynomal specal type of topologcal spaces. ) Study questons () to (v) of problem (11) for ths R. ) Enumerate all the specal features assocated wth ths R. 11. Let V = P(M[x] 1 ) = {collecton of all polynomal subsets of fnte order from M[x] 1 = { 1 a x / a Z 15 h ; x 11 = 1}} be the MOD specal dual lke number coeffcent polynomal subsets V o, V, V, V, V and V be the MOD specal dual lke number coeffcent polynomal subsets specal type of topologcal spaces. ) Prove o(v) <. ) ) Fnd all strong MOD specal dual lke number coeffcent polynomal specal type of topologcal subspaces. Prove V has no MOD subset polynomal specal type of topologcal deals.

109 18 MOD Natural Neutrosophc Subset Topologcal v) Fnd all MOD subset topologcal zero dvsor and nlpotents f any. v) Fnd all specal features assocated wth ths V. 11. Let W = P(N[x] 5 ) = {collecton of all polynomal subsets from N[x] 5 = { 5 a x / x = 1, a Z 5 g (or C (Z 1 ) or Z 1 h or Z 1 g)}} be the MOD natural neutrosophc polynomal subsets of fnte order. W o, W, W, W, W and W be the MOD natural neutrosophc polynomal subset specal type of topologcal spaces. Study questons () to (v) of problem (11) for ths W.

110 Chapter Three MOD SUBSET SPECAL TYPE OF NTERVAL TOPOLOGCAL SPACES n ths chapter we proceed onto defne mod specal type of nterval topologcal spaces usng [, n), [, n), [, n) g, C([, n)); C ([, n)), [, n)g, [, n) h, [, n) h, [, n) k and [, n) k. We wll llustrate ths stuaton by some examples and obtan all specal features assocated wth them. Example.1. Let S = {P([, )) be the power set of [, ). S s called mod nterval subsets. Defne on S the operaton +,, and. +. Clearly {S, +} s the mod nterval subset semgroup under {S, } be the mod nterval subset semgroup under. {S, } be the mod nterval subset semgroup under {S, } be the mod nterval subset semgroup under. We wll llustrate ths stuaton by some workng.

111 11 MOD Natural Neutrosophc Subset Topologcal Let A = {., 1.5,., 1.} and B ={1.5,.,.,.1} S. A B ={., 1.5,., 1.,.,.1} A B = {1.5,.) A B ={.8,.5,.5, 1.8,.9,.9,., 1.8,.,.1,.8,.,.15,.,.1} A + B = {.7,, 1.8,.7,.5, 1.8,., 1.5,., 1.9,.7, 1.,.1, 1.51,.1, 1.1} V Clearly,, and V are dstnct. So all the mod nterval subset semgroups are dstnct. Thus S o = {S,, }, S = {S, +, }, S = {S,, }, S = {S,, }, S = {S,+, } and S = {S, +, } be the sx dstnct mod subset nterval specal type of topologcal spaces. All these spaces enjoy specal propertes. S o, S, S, S and S has mod nterval subsets whch are topologcal dempotents. However S o, S and S does not contan zero dvsor, but the other three mod topologcal spaces contan zero dvsors; whch are defned as mod nterval subsets specal type topologcal space zero dvsors. Let A ={, } S. A A ={} s the mod nterval subset specal type of topologcal nlpotent subset of S. Apart from ths we are not n a poston to get mod nterval subset specal topologcal space zero dvsors.

112 MOD Subset Specal Type of nterval 111 However only the set B ={, 1} S s such that B B = {, 1}; whch we term as mod nterval trval specal topologcal dempotent of S, S and S = S. Example.. Let T ={P([, ) g)} = {collecton of all subsets from [, ) g = {a + bg /a, b [, ), g = }} be the mod nterval subset dual numbers. {T, +}, {T, }, {T, }and {T, } be the mod nterval dual number semgroups. T o = {T,, }, T = {T, +, }, T ={T, +, }, T = {T, +, }, T = {T,, } and T ={T,, } be the sx mod nterval dual number subset specal type of topologcal spaces. T, T and T = T have nfnte number of mod nterval subset specal topologcal space nlpotents and zero dvsors. Let A = {.g, g,.5g, 1.78g} B = {.g, 1.98g, g, 1.g, 5.789g} T. Clearly A andand nfact the reader s left wth the task of fndng mod nterval subset dual number specal type of topologcal subspaces. One can fnd easly deals but they are not strong. Further f P T o s an deal P need not be an deal of T, T, T, T and T.

113 11 MOD Natural Neutrosophc Subset Topologcal Smlarly f R T s an deal, R need not be an deal of T or T or T or T or T o. So obtanng mod nterval dual number subset specal type of topologcal strong deals s not possble. Only we can have strong subspaces. Example.. Let R = {P(C[, ))} be the power set of C([, )) = {a + b F / a, b [, ); F = 5}. R wll be known as mod nterval fnte complex number subsets. Defne on R, +,, and so that {R, +}, {R, }, {R, } and {R, } are mod nterval fnte complex number subset semgroups. All the four semgroups are dstnct. R o ={R,, }, R ={R, +, }, R = {R, +, }, R = {R, +, }, R = {R,, } and R = {R,, } are the mod fnte complex number nterval subset specal type of topologcal spaces all of whch are of nfnte cardnalty and are dstnct. Fndng subspaces, mod topologcal zero dvsors, mod topologcal nlpotents and mod topologcal deals are left as an exercse to the reader. Example.. Let T = {P([, 7) g)} be the mod nterval fnte dual number subset collecton. {T, +}, {T, }, {T, } and {T, } be the mod nterval fnte dual number subset semgroups, T o, T, T, T, T and T be the mod nterval fnte dual number subset specal type of topologcal spaces. All of them are dstnct and s of nfnte cardnalty. Further the mod topologcal nterval spaces T, T and T contan nfnte number of zero dvsors and nlpotents.

114 MOD Subset Specal Type of nterval 11 Ths has mod strong subspaces of both fnte and nfnte order whch are not mod topologcal deals. Consder B = {P(Z 17 )}; B o, B, B, B, B and B are mod subset subspaces; so B s a strong mod subset topologcal subspaces of T. D = {P(Z 17 g)} s agan a strong mod topologcal subset subspace of T. Both B and D are of fnte order and they are not deals of T. E ={P([, 17)g)} T s agan a mod subset strong topologcal subspace of T whch s not an deal of T. Smlarly F = {P([, 17))} T s a mod subset strong topologcal subspace of E whch s not an deal of T. Thus both E and F are of nfnte cardnalty. Further we once agan just recall a mod subset topologcal subspace L o T o to be an deal f for all subsets of L o and for all subsets W of T o. W L o and W L o. W of T Lkewse for L T s an deal f for all of L and W + L and W L. So fndng mod subset nterval topologcal deals happens to be a very dffcult problem. Smlar defnton n case of T, T, T and T.

115 11 MOD Natural Neutrosophc Subset Topologcal When T s an deal for all the topologcal spaces we call to be mod strong subset nterval topologcal deal of T o, T, T,, T. Study n ths drecton s challengng for there s no mod subset nterval strong specal type of topologcal deals. However there are mod subset nterval strong specal type of topologcal subspaces assocated wth T. Example.5. Let A = {P( [, ))} = {collecton of all subsets from [, ) = {a + where t s a nlpotent or zero dvsor or [,) t dempotent from Z ; a [, )}} be the mod natural neutrosophc nterval subset collecton. {A, +}, {A, }, {A, and be the mod natural neutrosophc modulo nteger nterval subset semgroups. All the four semgroups are dstnct. Further all these semgroups are of nfnte order havng both fnte and nfnte order subsemgroups. But all deals A are of nfnte order {A, } can have zero dvsors and nlpotents {A, +} can have dempotents. Every element n {A, } and {A, } are dempotents. We can have n {A, } deals of fnte order. A o = {A,, }, A A = {A, +, }, A = {A, +, }, = {A, +, }, A ={A,, } and A ={A,, } be the mod subset nterval natural neutrosophc modulo nteger specal type of topologcal spaces. All these spaces are of nfnte order and are dstnct.

116 MOD Subset Specal Type of nterval 115 Studyng compactness, connectedness etc; happens to be challengng problem. Let X = { + + +,, + } A., +,, 1,} and Y = { +, 1,, We fnd, X Y = { }, +, +,, 1,,, X Y = {, 1, + } X + Y = {, 5 + +, + + +, 1 +, +, +, +, 1 + +, 5,, 1, , +, + + +, + +, 5 + +, + +, + +, + +, +, +, 1 +, } X Y = { +, +, +,, 1, } {, 1, +, + +, } = {,,, +, +, +, +,, 1, +, +, + +, + +, + + +, + +, + +, +, +, + } V We have used natural neutrosophc zero domnated product and not the usual zero domnated product. The four equatons are dstnct hence the sx topologes are also dstnct. reader. Fndng subspaces deals etc are left as exercse to the A has mod natural neutrosophc specal topologcal dempotents subsets wth respect to + and also.

117 11 MOD Natural Neutrosophc Subset Topologcal Example.. Let B = {P([, 9) )} = {collecton of all natural neutrosophc nterval dual number subset collecton from [, 9) g ={a + bg + / a, b [, 9), g =, t Z 9 g t g s a zero dvsor or dempotent or nlpotent} be the mod natural neutrosophc nterval dual number subset collecton. {B, }, {B, }, {B, } and {B, +} be the four dstnct mod natural neutrosophc nterval dual number subset semgroups and B o, B, B, B, B and B be the mod natural neutrosophc nterval dual number subset specal type of topologcal spaces assocated wth these mod subset semgroups. Ths has nfnte number of mod natural neutrosophc dual number nterval subset specal type of topologcal zero dvsors and nlpotents under the operaton. However there are only a fnte collecton of mod nterval topologcal space dempotent subsets under. Study n ths drecton s a matter of routne so left as an exercse to the reader. These mod spaces has both of fnte order as well as of nfnte order, mod topologcal subspaces. Example.7. Let B = {P([, 8) h )} = {collecton of all mod natural neutrosophc specal dual lke number nterval subsets from [, 8) h = {a + bh + / a, b [, 8), h = h; t Z 8 h s such that t s a zero dvsor or nlpotent or an dempotent}. {B, +}, {B, }, {B, } and {B, } be the mod natural neutrosophc nterval specal dual lke number subset semgroup. Let B o, B, B, B, B and B be the mod subset nterval natural neutrosophc specal type of topologcal spaces bult usng these semgroups on B. h t

118 MOD Subset Specal Type of nterval 117 All propertes assocated wth B can be obtaned usng the fact h = h; ths task s left as an exercse to the reader. Example.8. Let C = {P(C ([, ))} be the collecton of all mod natural neutrosophc fnte complex number subset collecton. {C, +}, {C, }, {C, } and {C, } are the mod natural neutrosophc fnte complex number subset semgroup. C o, C, C, C, C and C are the sx dstnct mod specal type of topologcal spaces assocated wth C. All propertes assocated wth these mod semgroups and mod topologcal spaces s left as an exercse to the reader. Example.9. Let D = {P([, ) ) = {collecton of all subsets from the mod natural neutrosophc neutrosophc nterval [,) = {a + b + t / a, b [, ), =, t Z }} be the mod natural neutrosophc neutrosophc nterval subset collecton, {D, +}, {D, }, {D, } and {D, } be the mod natural neutrosophc neutrosophc subset nterval semgroups and D o, D, D, D, D and D are the mod natural neutrosophc neutrosophc subset nterval specal type of topologcal spaces assocated wth D. Study of strong subspaces of fnte and nfnte order s nterestng. However all these spaces have both fnte and nfnte order strong subspaces but no strong deals. n vew of all these we gve the followng theorem. Theorem.1. Let S = P([, n)) (or P( [, n)) or P([, n) ) or P([, n) ) or P(C([, n))) or P(C ([, n))) or P([, n) g) or P([, n) g ), P([, n)h) or P([, n)h )) or P([, n)k)) or P([, n)k )) be the mod nterval subset

119 118 MOD Natural Neutrosophc Subset Topologcal collectons (or mod natural neutrosophc subset collectons, approprately assocated wth these sets). ) S s of nfnte order. ) {S, +}, {S, }, {S, } and {S, } are mod nterval subset semgroups of nfnte order. ) These semgroups contan strong subsemgroups of fnte and nfnte order. v) These semgroup do not contan strong deals. v) These semgroups contan mod subset nterval zero dvsors, dempotents and nlpotents dependng on the sets chosen and n chosen. v) v) v) x) All mod subset nterval specal type of topologcal spaces S o, S, S, S, S and S are of nfnte order and are dstnct. These mod subset specal type of topologcal spaces has strong mod subset nterval subspaces of both fnte and nfnte order. S has no mod nterval subset topologcal deals. All mod topologcal nterval subset deals of S are of nfnte order. x) S, S and S has mod subset nterval topologcal zero dvsors, dempotents and nlpotents dependng of the subsets collecton used and on n. x) S, S, S ;, S o have dempotents wth respect +,, and (+ and depend on the set). The proof s drect and hence left as an exercse to the reader.

120 MOD Subset Specal Type of nterval 119 Next we proceed onto descrbe mod nterval subset matrx specal type of topologcal spaces and mod nterval matrx subset specal type of topologcal spaces and the mod natural neutrosophc analogues by examples. a1 a Example.1. Let W = { a / a P([, )); 1 5} be a a 5 the mod nterval subset matrx collecton. {W, +}, {W, }, {W, } and {W, n } be the mod nterval subset matrx semgroups. W o, W, W, n W, n W and n W be the mod nterval subset matrx specal type of topologcal spaces. Fndng mod strong topologcal subspaces of both fnte and nfnte order s a matter of routne so left as an exercse to the reader. n W, n W and W has nfnte number of mod n nterval subset matrx topologcal zero dvsors, W, W, respect to and. respect to n. n W, n W and W o has dempotents wth n W, n W and W n has dempotents wth W has both fnte and nfnte order strong mod nterval subset matrx specal type of topologcal subspaces, however strong deals of mod nterval subset matrx specal type of topologcal space does not exst.

121 1 MOD Natural Neutrosophc Subset Topologcal a1 a a a Example.11. Let V ={ a5 a a 7 a 8 / a P([, ) a9 a1 a11 a 1 g); 1 1, g = } be the mod nterval dual number subset matrx collecton. {V, n }, {V, }, {V, } and {V, +} be the mod nterval dual number subset matrx semgroups. V o, V, V, n V, n V and n V number subset matrx specal type of topologcal spaces. be mod nterval dual V, n n V and n V has nfnte topologcal zero dvsors. V o, V and V has every subset matrx to be an dempotent under or (and). V has strong mod dual number nterval subset matrx specal type of topologcal subspaces of both fnte and nfnte order, however has no strong mod dual number nterval subset matrx specal type of topologcal deals. a1 a a Example.1. Let Y = { a a5 a / a P(C[, 8)); 1 a7 a8 a 9 9} be mod nterval fnte complex number subset matrx collecton, {Y, +}, {Y, n }, {Y, },{Y, } and {Y, } be the mod nterval fnte complex number subset matrx semgroups. Let Y o, Y, Y, Y, n Y, Y, Y, n Y, n Y and Y n be the mod nterval fnte complex number subset matrx specal type of topologcal spaces assocated wth these fve mod semgroups. We see Y, Y, Y and Y are mod non n commutatve nterval fnte complex number subset matrx specal topologcal space.

122 MOD Subset Specal Type of nterval 11 The reader s left wth the task of fndng mod nterval fnte complex number subset matrx topologcal rght deals whch s not left deals and vce versa. These spaces, Y, Y, Y and n Y can have mod nterval fnte complex number subset matrx specal type of topologcal rght zero dvsors whch are not left zero dvsors and vce versa. The study n ths drecton s a matter of routne so left as exercse to the reader. a1 a a a Example.1. Let Z ={ / a P( [, 5)); 1 a5 a a 7 a8 8} be the mod nterval natural neutrosophc matrx wth subset entres collecton. {Z, +}, {Z, }, {Z, } and {Z, n } be the mod nterval natural neutrosophc subset matrx semgroups. {Z, +} has dempotents. Further n {Z, } and {Z, } every element s an dempotent. {Z, n } also has nlpotents and dempotents. We see the mod natural neutrosophc subset matrx specal type of topologcal spaces has mod natural neutrosophc subset matrx specal type of topologcal dempotents, nlpotents and zero dvsors. Next we proceed onto descrbe mod nterval natural neutrosophc fnte complex number matrx wth subset entres by some examples.

123 1 MOD Natural Neutrosophc Subset Topologcal a1 a a a Example.1. Let S ={ a5 a / a P(C ([, )); 1 a7 a8 a9 a 1 1} be the mod nterval natural neutrosophc fnte complex number matrces wth subset entres collecton. {S, +}, {S, n }, {S, } and {S, } are mod nterval natural neutrosophc fnte complex number matrces wth subset entres semgroup. A = {. F, 1, F } {, 1 F} { F,,} {1, F, 1} {,1 F,1} {1,,5 F} { F,, 8 F} {,5} { F, F} {.1,.8} and B = {. F,1,} {, 1} {5,,9 F} {,1, F} {1 F,1} {, F} {,5 F,} {, F,1} {1, 5 F} {.1,.5} S. A B = {. F, 1, F, } {,1 F,,1} {,,5,9 F, F} {1,1,, F} {1 F,1,1,} {1,,5 F,, F} {, F,,5 F, 8 F} {,5,1,, F} { F, F,5 F,1} {.1,.5,.8}

124 MOD Subset Specal Type of nterval 1 A B = {. F,1} { } {} {1, F} {1 F} { } {} { } { } {.1} A B = {. F,1, F,,.78,.9 F 5., {,,1 F}. F,F } {,1, 5 F,, {, 1, F,1, 8 F,18 F,F } F,} {1,1 F,9, 8,8 F,1 F} {, F,,1 F9} {,7 F, 1 F,, 1F 9} { F, F,8, } {.1,.8,.5,.} Fnally we fnd A + B; A + B = {1. F,,F, {,1 F,. F,,5 F, 1,11 F}. F,1. F,. F} {5,,9 F,7,, 9 F {1, F,1,,11, 9 F, F, 1 F} 1 F, F,1 F} {,1 1 F,11, {, F, 1,1 F 1 F, F.11 F} 5F,5 F} { F, 8 F,9 F,5 F, {,5, F,5 F 1 F, F,, 8 F} 1,15} {1 F,1 F, F,5 F} {.,.9,.,1.} V

125 1 MOD Natural Neutrosophc Subset Topologcal Thus all the four mod nterval subset matrx entres semgroups are dfferent. Hence the related mod natural neutrosophc nterval subset entres matrx specal type of topologcal spaces are also dstnct. These spaces has mod topologcal dempotents wth result and +. However fndng mod topologcal dempotents happens to be a dffcult task. a1 a Example.15. Let M ={ a a /a P ([, ) g ); 1 } be the mod nterval natural neutrosophc dual number matrces wth subset entres collecton {M, +}, {M, }, {M, }, {M, n } and {M, } be the mod nterval natural neutrosophc dual number matrces wth subset semgroups. M o, M, M, M, n M, M, M, n M, n M and n M be the mod nterval natural neutrosophc dual number matrx subset specal type of topologcal spaces. We wll show how the two products and n are dfferent. {g, 5g,1} {8g 1,} Let A = {, g, } {1g, g } and B = { 1g, g,,1} {1g, g } {,, g 1} {1g,8g} belong to M.

126 MOD Subset Specal Type of nterval 15 A B = {1,g,5g,,g,15g,,g,1g,1 g} {,1,g 1} {,18 1g,9,,18g, 1,1g,1g 18 g } {,1g, 18g 18,1g, 1g} {8g,,8,1 1g} {} {,8g} {} = {1,g,5g,,g, 15g,,g,1g,1 g, 1,1 g,5g 1,15,g 1, 18 15g,1 1g,g 1, 1g 1, 5g 1, g 15, g 1,1118,g 1, 8g 1, g} {,18 1g,9,,18g, 1,1g,1g 18, g,18,1g 9, 1g,g,1 1g 18,1g, 1g, 15 1g,9 1g, g,18 1g, 1 1g} {8g,,8 1 1g} {,8g} verfy. Clearly A n B A B. t s left for the reader to Thus M has mod topologcal zero dvsors, nlpotents under n and.

127 1 MOD Natural Neutrosophc Subset Topologcal Also M has mod topologcal rght zero dvsors whch are not left zero dvsors and vce versa under the product. So the mod topologcal spaces M, M, M and M n are non commutatve and {M, } s a mod non commutatve semgroup whch contrbute to t. a1 a Example.1. Let N = a a a 5 / a P ([,) h ; 1 5} be the mod nterval natural neutrosophc specal dual lke number matrx wth entres as subsets collecton. {N, +}, {N, n }, {N, } and {N, } be the mod nterval natural neutrosophc specal dual lke number matrx subsets semgroups. N o, N, N, n N, n N and n N be the mod nterval natural neutrosophc matrx subsets specal type of topologcal spaces. N has mod topologcal dempotents under all the four operatons. a1 a a a Example.17. Let P ={ a5 a a 7 a 8 / a P ([,) a9 a1 a11 a 1 k ; 1 1} be the mod nterval natural neutrosophc matrx subsets specal quas dual number collecton {P, n }, {P, }, {P, } and {P, +} be the mod nterval natural neutrosophc specal quas dual number matrx subset semgroups.

128 MOD Subset Specal Type of nterval 17 P o, P, P, P, n P and n n P be the mod nterval natural neutrosophc specal quas dual number matrx subset specal type of topologcal spaces. Fndng nontrval mod topologcal dempotents, nlpotents are zero dvsors are left as an exercse to the reader. a1 a a a 7 Example.18. Let Q = { a a 8 / a P([,8 ); 1 a a9 a5 a 1 1} be the mod nterval natural neutrosophc-neutrosophc subset matrx collecton. {Q, +}, {Q, n }, {Q, } and {Q, } be the mod nterval natural neutrosophc-neutrosophc subset matrx semgroup and Q o, Q, Q, Q, n Q and n be the mod n nterval natural neutrosophc-neutrosophc subset matrx topologcal spaces. Q reader. Studyng ther propertes s left as an exercse to the Next we proceed onto prove the followng result. Theorem.. Let M ={A =(a j ) m n / a j P([, p)) (or P( [, p)) or P(C[, t)) or P(C [, t)) or P([, s) ) or P([, s) ) or P([, a) g) or P([, a) g ) or P([, b) h) or P([, b) h ) or P([, c) k) or P([, c) k ) where p, t, s, a, b and c Z + ), 1 m, 1 j n; m, n < } be the mod nterval subset entres matrx collecton (mod nterval natural neutrosophc subset entres matrx collecton and so on).

129 18 MOD Natural Neutrosophc Subset Topologcal ) {M, +}, {M, }, {M, n } and {M, } are the mod nterval matrx wth subset entres semgroups. ) M o, M, M, M, n n M and n M are mod nterval subset entres matrx specal type of topologcal spaces. ) There are mod nterval subset matrx topologcal zero dvsors and nlpotents for approprate values of the nterval. v) M has mod nterval subset matrx strong topologcal spaces. v) M has no mod nterval subset topologcal strong deals. reader. Proof s drect and hence left as an exercse to the Next we proceed onto descrbe usng mod nterval matrx subset varous topologcal spaces by examples. a1 a Example.19. Let A ={P(M) where M = { a / a [, ); a a 5 1 5}} be the mod nterval matrx subset collecton {A, }, {A, }, {A, +} and {A, n } are mod nterval matrx subsets semgroup. A o, A, A, n A, n A and n A are mod nterval matrx subset specal type of topologcal spaces.

130 MOD Subset Specal Type of nterval 19 Let x = { 1, 1 5 1, } and y = { 1, 5, } A. x y = { } x y = { 1, 1 5 1, 1 1 8, 1, 5 } x + y ={ 1 5, 1 9, 1 8, 8, , , 1 1, 1, }

131 1 MOD Natural Neutrosophc Subset Topologcal 9 x n y ={, 15,, 5, 8, 1 1 1, 5 1 1,, 1 } V 8 We see all the four operatons are dstnct so the mod nterval topologcal spaces are also dstnct M s of nfnte order M has mod strong subspaces of both fnte and nfnte order. However all deals are of nfnte order and none of them are strong deals. Example.: Let N = P (D) = {collecton of all subsets from a1 a a a a 5 a D = { / a [, 1); 1}} be the mod a 7 a8 a 9 a1 a11 a1 nterval natural neutrosophc matrx subset collecton. {N, +}, {N, }, {N, } and {N, n } be the mod nterval natural neutrosophc matrx subset semgroups. N o, N, N, n N, n N and n N be the mod nterval natural neutrosophc matrx specal type of topologcal spaces. deals. Ths has mod strong subspaces and has no mod strong

132 MOD Subset Specal Type of nterval 11 n N, n N and N n and dempotents. However nlpotents. have mod topologcal zero dvsors n N, Example.1. Let V = {P(W) where n N and N n has no nontrval a1 a a a a 5 a a 7 a 8 W = { / a C([, )); F =, 1 a9 a1 a11 a 1 a1 a1 a15 a1 1} be the mod nterval fnte complex number matrx subset collecton. {V, +}, {V, n }, {V, }, {V, } and {V, } be the mod nterval fnte complex number matrx subset semgroups. Clearly {V, } s the only non-commutatve structure V o, V, V, n V, V, n V, n V, n V, V and V be the mod nterval fnte complex number matrx subset specal type of topologcal spaces assocated wth V. Clearly V, V, V and V n are all non-commutatve mod nterval fnte complex number matrx subset topologcal spaces assocated wth V. All propertes assocated wth these mod nterval topologcal spaces s consdered as a matter of routne so leftas exercse to the reader.

133 1 MOD Natural Neutrosophc Subset Topologcal a1 a a a a5 a a 7 a8 a 9 Example.. Let M ={P(T) where T = { / a a1 a11 a1 a1 a1 a 15 a1 a17 a18 C ([, 1)); 1 18} be the mod nterval natural neutrosophc fnte complex number matrx subset collecton {T, +}, {T, }, {T, }and {T, n } be the mod nterval natural neutrosophc fnte complex number subset semgroups. T o, T, T, n T, T and T be the mod nterval natural neutrosophc fnte complex number matrx subset specal type of topologcal spaces assocated wth these four semgroups. Study of propertes assocated wth T s a matter of routne so left as exercse to the reader. a1 a a Example.. Let B = {P(L) / L = { / a [, ) ; a a 5 a 1 }} be the mod nterval neutrosophc matrx subset collecton. {B, +}, {B, }, {B, } and {B, n } are the mod nterval neutrosophc matrx subset semgroups. B o, B, B, n B, n B and n B are the mod nterval neutrosophc matrx subset specal type of topologcal spaces assocated wth the semgroup.

134 MOD Subset Specal Type of nterval 1 a1 a D = {P(S) / S = { / a [, ); 1 }} be the a mod nterval neutrosophc matrx subset collecton D B s a mod nterval neutrosophc strong topologcal subspaces of B however D s not a strong deal of B. Gettng other propertes related wth B s a matter of routne so left as an exercse to the reader. a1 a a a a 5 a a 7 a 8 Example.. C = {P(N) / N = { / a a9 a1 a11 a 1 a1 a1 a15 a1 [, ; 1 1}} be the mod nterval natural neutrosophc-neutrosophc matrx subset collecton. {C, +}, {C, }, {C, }, {C, n } and {C, } be the mod nterval natural neutrosophc-neutrosophc matrx subset semgroup. Let C o, C, C, C, n C, C, C, n C, n C and n C be the mod natural neutrosophc-neutrosophc matrx subset specal type of topologcal spaces. Clearly C, n C, C and C are mod nterval natural neutrosophc-neutrosophc subset matrx non-commutatve specal type of topologcal spaces. Fnd mod strong subspaces and show C has no mod strong deals. Example.5. Let F = {P(Z) / Z =

135 1 MOD Natural Neutrosophc Subset Topologcal a1 a a a a 5 a { / a [,1) g; 1 1} a 7 a8 a9 a1 a11 a1 be the mod nterval dual number matrx subset collecton. {F, +}, {F, }, {F, } and {F, n } be the mod nterval dual number matrx subset semgroup. F o, F, F, F, n n F and n F be the mod nterval dual number matrx subset specal type of topologcal spaces. n These spaces F, n F and n F have mod nterval dual number topologcal nlpotents and zero dvsors n nfnte order. Further F has mod nterval dual number subset matrx topologcal subspaces P whch are such that P n P = { }. Ths s the specal feature assocated manly wth mod nterval dual number matrx subset specal type of topologcal spaces. a1 a Example.. Let B ={P(Y) / Y = { a / a [, 1) g ; 1 a a 5 5}} be the mod nterval natural neutrosophc dual number matrx subset collecton. {Y, +}, {Y, }, {Y, } and {Y, n } be the mod nterval natural neutrosophc dual number matrx subset semgroups.

136 MOD Subset Specal Type of nterval 15 Y o, Y, Y, n Y, n Y and n Y be the mod nterval natural neutrosophc dual number matrx subset specal type of topologcal spaces assocated wth these semgroups. Y o, Y, Y, n Y, n Y and n Y be the mod nterval natural neutrosophc dual number matrx subset specal topologcal spaces assocated wth these four semgroups. Y has mod nterval natural neutrosophc dual number matrx subset strong topologcal subspaces and some of them are such that then square s zero. We have the followng result. Theorem.. Let S = {P(B) / B = {M = (m j ) / m j [, n) g (or [, n) g ); 1 t, 1 j s}} be the mod nterval dual number (or mod nterval natural neutrosophc dual number) matrx subsets collecton. {S, +}, {S, n }, {S, } (when t = s), {S,} and {S, } be the mod nterval dual number (or mod nterval natural neutrosophc dual number) matrx subset semgroups. S o, S, S, S, (or/and S, S, S, n S ) n S n n and S be the mod nterval natural neutrosophc dual number specal type of topologcal spaces accordng as s = t or s t. ) o(s) =. ) ) v) S has fnte order mod nterval matrx subset topologcal spaces whch are strong. S has strong mod nterval matrx subset topologcal spaces whch are of nfnte order. S has mod nterval matrx subset topologcal spaces has zero dvsors, nlpotents and dempotents for approprate n of [, n). v) All mod nterval matrx subset topologcal deals of S are of nfnte order.

137 1 MOD Natural Neutrosophc Subset Topologcal v) v) S has no mod strong topologcal deals. S has mod topologcal subspaces M of both fnte and nfnte order such that M n M ={()}, M M = {()}. reader. Proof s drect and hence left as an exercse to the The seventh property s only a specal feature enjoyed by mod nterval dual numbers and mod natural neutrosophc nterval dual numbers. a1 a a a Example.7. Let W ={P(Z) / {Z = { a5 a / a [, ) a7 a8 a9 a 1 h; 1 1}} be the mod nterval specal dual lke number matrx subset collecton. {W, +}, {W, }, {W, } and {W, n } be the mod nterval specal dual lke number matrx subset semgroups W o, W, W, W, n W and n be the mod nterval specal dual n W lke number matrx subset specal type of topologcal spaces usng the mod semgroups. W has mod nterval matrx subset specal dual lke number strong specal type of topologcal subspaces of both fnte and nfnte order. We have mod nterval specal dual lke number matrx subset topologcal zero dvsors, dempotents and nlpotents.

138 MOD Subset Specal Type of nterval 17 Example.8. Let F = {P(T) / T ={collecton of all matrces a1 a a a a5 a / a [, 8) h ; 1 9}} be the mod a7 a8 a 9 nterval natural neutrosophc specal dual lke number matrx subset collecton {F, +}, {F, }, {F, } and {F, n } be the mod nterval natural neutrosophc specal dual lke number matrx subset semgroups. F o, F, F, n F, F, n F, n F, n F, F and F be the mod nterval natural neutrosophc specal lke number matrx subset specal type of topologcal spaces G = {P(H) / a1 a a H = a a5 a / a Z 8, 1 9} be the mod nterval a7 a8 a 9 specal dual lke number subset matrx specal type of topologcal subspaces whch s strong and s of fnte order. Replace H n G by K where entres are from Z 8 H. Then ths K s also a mod strong topologcal subspace and so on. Example.9. Let L ={P(W) / a1 a a a a5 a a 7 W = { a8 a9 a1 a11 a1 a1 a / a [, ) k ; 1 1 1}} be the mod nterval specal quas dual lke matrx subset collecton. t s matter of routne to defne mod nterval semgroups and mod nterval subset specal topologcal spaces and study the propertes related wth them.

139 18 MOD Natural Neutrosophc Subset Topologcal a1 a Example.. Let Z ={P(M) / M = { / a [, ) k ; 1 a a }} be the mod nterval natural neutrosophc specal quas dual number matrx subset collecton. {Z,+}, {Z, n }, {Z, } and {Z, } be the mod nterval natural neutrosophc specal quas dual number matrx subset semgroups Z o, Z, Z, Z, n Z and n be the mod natural n neutrosophc nterval specal quas dual number matrx subset topologcal space. a1 a Y = {P(N) / N ={ / a [, )k; 1 } Z be a a the mod natural neutrosophc nterval specal quas dual number specal type of matrx subset topologcal subspaces Y s a strong mod nterval topologcal subspace of nfnte order whch s not a strong topologcal deal but not even a mod nterval topologcal deal. Z reader. Study n ths drecton s left as an exercse to the Now we proceed onto descrbe mod nterval subset specal type of polynomal topologcal spaces by examples. Example.1. Let W = M[x] = { a x / a P([, 1))} be the mod nterval polynomal wth subset coeffcents collecton {M[x], }, {M[x], }, {M[x], } and {M[x], +} be the mod nterval polynomal subset coeffcent semgroup.

140 MOD Subset Specal Type of nterval 19 Let p(x) = {,.5,.1}x + {.7,.}x + {.5,,.1} and q(x) = {.1,.5,, }x + {, 1}x + {,.1, 7} M[x] p(x) + q(x) = {.1,.,.,5.5, 5,., 9, 8.5,.1,,.1}x + {.7,., 1.7, 1.}x + {.5,,.1,.,., 7.5, 9, 7.1} p(x) q(x) = {,.1,.5}x + {,.1} p(x) q(x) = {,.5,.1, }x + {.7,.,, 1}x + {.1,,.5, 7} p(x) q(x) = {.,.5,.1, 7.5, 5,.5,.5, 8,,., 9}x + {.7,., 1.75,.5,., 1.,.1,.}x + {.5,.,.1, 1.5, 5,.5,,,., 1.5,,.}x + {, 5,.,,.5,.}x +{1.,.,.7,.}x + {1,,.,.5,,.1}x + {1,,.,.5,.1,.5,,.7} + {1.,.,.7,.,.9}x = {.,.5,.1, 7.5, 5,.5,.5, 8,., 9}x +{.7,., 7.75,.5,., 7., 8.1,., 5.7, 5.,.75, 5.5, 9.,., 7.1, 5.,.7,., 1.95,.7,., 1.,.,.8,.7,.,.75,.5, 7.,., 5.1,.,.57,.5,.5,,.7,.7,.,.1,.7,.,.5,.8,.5, 1.5,.,.9}x + {.5,.,.1, 1.5, 5,.5,,,., 1.5,,.}x + {1.,.,.7,.}x + {., 5., 1., 1.9,., 1.5, 1.,.,.,.9,.,.5, 1.7,.7,.7,.57,.7,.17, 1.,.,.,.5,.,.1, 5.9, 8.9, 5.1, 5.,.9, 5}x + {1,,.,.5,.1,.5,,.7} V All the four mod nterval subset polynomal coeffcent semgroups are dstnct. Hence W o, W, W, W, W and W are the sx dstnct mod nterval subset coeffcent polynomal specal type of topologcal spaces are dstnct.

141 1 MOD Natural Neutrosophc Subset Topologcal Fndng strong mod topologcal spaces, mod topologcal deals and mod topologcal nlpotents and zero dvsors happens to be matter of routne. Example.. Let S = W[x] = { a x / a P{ [, 8))} be mod nterval natural neutrosophc subset coeffcent polynomal collecton. {S, +}, {S, o }, {S, }, {S, } and {S, } be the mod nterval natural neutrosophc subset coeffcent polynomal semgroups. Let S o, S, S, S, S, S, S, S, S and S be the mod natural neutrosophc nterval subset coeffcent polynomal specal type of topologcal spaces assocated wth the semgroups. Let p(x) = { , 8 +.,.111, 1 }x + { , 1.7,.79} S.,,.1 + }x 7 + { 8, , nterested reader can prove the two operaton and are dstnct on S. The mod nterval natural neutrosophc subset coeffcent topologcal zeros wth respect to and are dfferent. deals. There are mod strong subspaces whch are not strong For nstance L = V[x] = { a x / a P ([, 8)} S s a mod strong nterval subset coeffcent polynomal topologcal space but s not a mod strong nterval subset deal.

142 MOD Subset Specal Type of nterval 11 However L s an deal of S o alone. Example.. Let R = P[x] = { ax / a C ([, )); F = 19} be the mod nterval fnte complex number subset coeffcent polynomal collecton {R, +}, {R, }, {R, } and {R, } be the mod nterval fnte complex number subset coeffcent polynomal semgroups and R o, R, R, R, R and R the related mod nterval fnte complex number coeffcent topologcal spaces. All propertes can be derved related to R. All deal of R are of nfnte order and R has no mod nterval strong topologcal deals. Example.. Let T = {B[x]} = { ax / a P (C [, 1))} the mod nterval natural neutrosophc fnte complex number subset coeffcent polynomal collecton, {T, }, {T, o }, {T, +}, {T, } and {T, } be the mod nterval natural neutrosophc fnte complex number subset coeffcent polynomal semgroups. Let T o, T, T, T, n T, n T, T, T, n T and n T be the mod nterval natural neutrosophc fnte complex number subset coeffcent polynomal specal type of topologcal spaces assocated wth these 5 semgroups. All propertes can be derved wth approprate modfcaton. Example.5. Let E = {D[x]} = { a x / a P ([, ) } be the mod nterval neutrosophc subset coeffcent polynomal

143 1 MOD Natural Neutrosophc Subset Topologcal collecton. E o, E, E, E, E, E, E, E, E and E be the mod nterval specal type of topologcal spaces. W = B[x] = { a x / a P ([, ))} be the mod nterval neutrosophc subset coeffcent polynomal strong specal type of topologcal subspace. Clearly W s not a strong mod topologcal deal. Example.. Let H = {V[x]} = { a x / a P ([, 15) )} be the mod nterval natural neutrosophc-neutrosophc subset coeffcent polynomal collecton. As usual H o, H, H, H, H, H, H, H, H and H be the mod nterval natural neutrosophc-neutrosophc subset coeffcent polynomal specal type of topologcal spaces. H, H, H has mod fnte order topologcal subspaces. Under H, H, H, H, H and H are only of nfnte order. all subspaces Example.7. Let J = {W[x]} = ax / a P ([, ) g )} be the mod nterval natural neutrosophc dual number subset coeffcent polynomals collecton {J, +}, {J, o }, {J, }, {J, } and {J, } be the mod nterval natural neutrosophc dual number subset coeffcent polynomal semgroups. Assocated wth these 5 semgroups we have 1 dstnct mod nterval natural neutrosophc dual number subset coeffcent polynomal specal type topologcal spaces.

144 MOD Subset Specal Type of nterval 1 The man feature enjoyed by these spaces s we have mod nterval natural neutrosophc dual number subset coeffcent polynomals specal type of subspaces V such that V V = {} and V can be both of fnte or of nfnte order. There are strong mod nterval subset specal type of topologcal spaces however they are not mod strong topologcal deals. Example.8. Let M ={T[x]} = { a x / a P([, 1) g)} be the mod nterval dual number subset coeffcent polynomal collectons. {M, +}, {M, }, {M, } and {M, } be the mod nterval dual number subset coeffcent polynomal semgroups. Assocated wth these mod semgroups we have M o, M, M, M, M and M, the mod nterval dual number subset coeffcent polynomal specal type of topologcal spaces. W 1 = {L[x]} = { ax / a P (Z 1 )}, W = {P[x]} = { a x / a P (Z 1 g)} and W = {R[x]} = { a x / a P (Z 1 g)} be the mod nterval dual number subset coeffcents polynomal topologcal subspaces. All these are strong mod topologcal subspaces. None of them are deals. We see W W = {}.

145 1 MOD Natural Neutrosophc Subset Topologcal Example.9. Let K = {L[x]} = { a x / a P ([, 1) h), h = h} be the mod nterval specal dual lke number subset coeffcent polynomal collecton. {K, +}, {K, }, {K, } and {K, } be the mod nterval specal dual lke number subset coeffcent polynomal semgroups. K o, K, K, K, K and K be the mod nterval specal dual lke number subset coeffcent polynomal specal type of topologcal spaces assocated wth these semgroups. T = {P[x]} = { a x / a [, 1)h} K s the mod nterval subset coeffcent polynomal strong topologcal subspace of K but s not a strong topologcal deal. Only T s an deal f we demand for every p(x) T and for every q(x) K; p(x) q(x) T and p(x) q(x) T. However p(x) q(x) T o, T, T, T. p(x) + q(x) T o, T, T and T. But f we restran t s suffcent f under only one of the operaton t should belong to T then except T all wll be deals for or the set T s an deal, however p(x) q(x) and p(x) + q(x) do not belong to T for any p(x) T and q(x) K. Hence the clam. Ths has mod topologcal zero dvsors but mod topologcal dempotents does not exst. Also ths has mod topologcal nlpotents as [, 1) has no nlpotents.

146 MOD Subset Specal Type of nterval 15 Example.. Let L = {M[x]} = ax / a P ([, ) h } be the mod nterval natural neutrosophc specal dual lke number subset coeffcent polynomal collecton {L, +}, {L, }, {L,, {L, o } and {L, } be the mod nterval natural neutrosophc specal dual lke number subset coeffcent polynomal semgroups. L o, L, L x, L, L, L, L, L x, L and L be the mod nterval natural neutrosophc specal dual lke number subset coeffcent polynomal specal type of topologcal spaces assocated wth the fve mod nterval semgroups. The study of propertes related wth L happens to be a matter of routne so left as an exercse to the reader. Example.1. Let M = {R[x]} = { a x / a P [, ) k; k = k} be the mod nterval specal quas dual number subset coeffcent polynomal collecton. {M, +}, {M, }, {M, } and {M, } be the mod nterval subset coeffcent semgroups. M o, M x, M, M, M and M be the mod nterval specal quas dual number subset coeffcent polynomal specal type of topologcal spaces. M has mod topologcal zero dvsors, only mod topologcal trval dempotents and mod topologcal nlpotents as s such that t contans zero dvsors and nlpotents. Example.. Let N = {S[x]} = { ax / a P ([, 7) k )} be the mod nterval natural neutrosophc specal quas dual number subset coeffcent polynomal collecton.

147 1 MOD Natural Neutrosophc Subset Topologcal {N, +}, {N, o }, {N, }, {N, } and {N, } be the mod nterval natural neutrosophc specal quas dual number subset coeffcent polynomal semgroups. N o, N, N, N, N, N, N, N, N and N be the mod nterval neutral neutrosophc specal quas dual number specal type of topologcal space of subset coeffcents polynomal. Study of the related propertes s a matter of routne so left as an exercse to the reader. Next we proceed onto descrbe by examples mod nterval polynomal subset specal type of topologcal structures. Example.. Let W = P(S[x]) = {collecton of all subsets from S[x] = { a x / a [, 1)}} be the mod nterval polynomal subset collecton, {W, +}, {W, }, {W, } and {W, } be the mod nterval polynomal subset semgroups. Let A = {.5x , 5x +.1x +.1,.5x +.5} and B = {.5x +.5, 7x + 5x +.5, 1x +.} W. A B = {.5x , 5x +.1 x +.1,.5x +.5, 7x + 5x +.5, 1x +.} A B = {.5x +.5} A + B = {.5x 8 +.5x , 5x +.1x +.5x +.1x +.1, 5x +.1,.5x + 7x + 5x , 5x + 7x + 5.1x +.,.5x 8 + 1x + 1.5, 9.5x + 5x +.55, 1.5x x + 1x +.1x +.}

148 MOD Subset Specal Type of nterval 17 A B = {1.5x 1 +.5x +.5x , 1.5x 5 +.5x +.5x +.5x +.5x +.5,.5x +.15x +.15x +.5,.5 x x +.5x x +.5x , 5x 5 +.7x +.7x + 5x +.5x +.5x +.5x +.5x +.5,.5x +.5x +.5x +.5x +.15x +.5x 1 + 9x +.x 8 +., x 5 + x +.1x +.x +.x +., x +.5x +.15x +.} V Clearly all the equatons,, and V are dstnct hence the mod nterval polynomal subset topologcal spaces assocated these four mod semgroups are dstnct. W o, W x, W, W, W and W has mod nterval polynomal subset strong topologcal spaces. No nontrval dempotents wth respect to. However under and every element of W s an dempotent. Example.. Let N = {P(W[x])} = {collecton of all subsets from W[x] = { a x / a [, )}} be the mod nterval natural neutrosophc polynomal subsets collecton. {N, +}, {N, }, {N, }, {N, } and {N, } be the mod nterval natural neutrosophc polynomal subsets semgroup. We see N o, N x, N, N, N, N, N, N, N and N are the mod nterval natural neutrosophc polynomal subset specal type of topologcal spaces assocated wth these 5 semgroups. Propertes related wth them can be desred wth approprate changes.

149 18 MOD Natural Neutrosophc Subset Topologcal Example.5. Let V = P(W[x]) = {collecton of all subsets from W[x] = { a x / a C ([, )}} be the mod nterval fnte complex number polynomal subset collecton, {V, }, {V, }, {V, +}and {V, } be the mod nterval fnte complex number polynomal subsets semgroup. V o, V, V, V, V and V be ther respectve mod nterval fnte complex number polynomal subset specal type of topologcal spaces. All related propertes of V can be desred a routne way wth approprate changes. Example.. Let Z = {P(V[x])} = {collecton of all subsets from V[x] = { a x / a C ([, 7))} be the mod nterval natural neutrosophc fnte complex modulo nteger polynomal subset collecton. {Z, +}, {Z, o }, {Z, }, {Z, } and {Z, } be the mod nterval natural neutrosophc fnte complex number polynomal subset semgroups. Z o, Z, Z, Z, Z, Z, Z, Z, Z and Z be the mod nterval natural neutrosophc fnte complex number polynomal subset specal type of topologcal spaces assocated wth these 5 semgroups. t s left as an exercse for the reader to derve all the related propertes.

150 MOD Subset Specal Type of nterval 19 Example.7. Let Y = {P(T[x])} = {collecton of all subsets from T[x] = { a x neutrosophc polynomal subset collecton. / a [, 9) }} be the mod nterval We see {Y, +}, {Y, }, {Y, } and {Y, } be the mod nterval neutrosophc polynomal subset semgroups. Y o, Y, Y, Y, Y and Y be the mod nterval neutrosophc subset polynomal specal type of topologcal spaces assocated wth the semgroups. P 1 = P(R[x]) = {collecton of all subsets from R[x] = { ax / a [, 9)}}, a x P = P(R[x]) = {collecton of all subsets from R 1 [x] = / a [, 9)}} Y are both mod nterval neutrosophc polynomal subset strong topologcal subspaces that are not strong topologcal deals. However R 1 [x] s an deal for some of the topologcal spaces lke T. Study n ths drecton s nnovatve and nterestng. Example.8. Let R = {P(L[x])} = {collecton of all subsets from L[x] = { a x / a [, 1) }} be the mod natural neutrosophc-neutrosophc polynomal subset collecton. {R, +}, {R, }, {R, }, {R, } and {R, n } be the mod nterval natural neutrosophc-neutrosophc polynomal subset semgroup.

151 15 MOD Natural Neutrosophc Subset Topologcal R o, R, R, R, R, R, R, R, R and R be the mod nterval natural neutrosophc-neutrosophc polynomal subset specal type of topologcal spaces. All propertes of these mod specal spaces can be derved wth approprate modfcatons. Example.9. Let M = {P(B[x])} = {collecton all subsets from B[x] = { a x dual number subset polynomal collecton. / a [, ) g}} be the mod nterval {M, +}, {M, }, {M, } and {M, } be the mod nterval dual number polynomal subset semgroups. M o, M, M, M, M and M be the mod nterval dual number subset polynomal specal type of topologcal spaces. Ths mod topologcal space has nlpotents and zero dvsors. nfact M has mod topologcal subspaces say S such that S S = {}. Example.5. Let B = {P(D[x])} = {collecton of all subsets from D[x] = { a x / a [, 1) g }} be the mod nterval natural neutrosophc dual number polynomal subset collecton. {B, +}, {B, }, {B, }, {B, } and {B, } be the mod nterval natural neutrosophc dual number polynomal subset semgroup. B o, B, B, B, B, B, B, B, B and B be the mod nterval natural neutrosophc dual number polynomal subset specal type of topologcal spaces assocated wth ths fve mod nterval semgroups.

152 MOD Subset Specal Type of nterval 151 Only these spaces have both mod nterval natural neutrosophc zero dvsors and nlpotents, mxed zero dvsors and nlpotents and mod nterval usual zero dvsors and nlpotent. Ths mod dual number spaces behaves dfferently and dstnct from other so study of these happens to be both nnovatve and nterestng. Example.51. Let S = {P(R[x]) = {collecton of all subsets from R[x] = { a x / a [, 1) h}} be the mod nterval specal dual lke number polynomal subset collecton {S, +}, {S, }, {S,} and {S, } be the mod nterval specal dual lke number polynomal subset semgroups. Let S o, S, S,S, S and S be the mod nterval specal dual lke number polynomal subset specal type of topologcal spaces. { a x T = {P(M[x])} = {collecton of all subsets from M[x] = / a [, 1)h}} be the mod nterval specal dual lke number subset specal type of strong topologcal subspaces. Clearly T s not a mod strong deal of S. Study n ths drecton s also nnovatve and left as an exercse to the reader. Example.5. Let W = {P(T[x]) = {collecton of all subsets from T[x] = { a x / a [, 17) h }} be the mod nterval natural neutrosophc specal quas dual lke number polynomal subsets collecton, {W, +}, {W, }, {W, }, {W, } and

153 15 MOD Natural Neutrosophc Subset Topologcal {W,x} be the mod nterval natural neutrosophc specal dual lke number polynomal subset semgroups. W o, W, W, W, W, W, W, W, W and W be the mod nterval natural neutrosophc specal dual lke number polynomal subset specal type of topologcal spaces bult usng these 5 mod nterval semgroups. Ths W has strong mod nterval topologcal subspaces but has no mod nterval topologcal deals. Study n ths drecton s nterestng and ths work s left as an exercse to the reader. Example.5. Let M = {P(C[x])} = {collecton of all subsets from C[x] = a x / a [, ) k}} be the mod nterval specal quas dual number polynomal subsets collecton {M, +}, {M, }, {M, } and {M, } be the mod nterval specal quas dual number polynomal subset semgroups. M o, M M, M, M and M be the mod nterval specal quas dual number polynomal subset specal type of topologcal spaces assocated wth these semgroups. { a x N = {P(V[x])} = {collecton of all subsets from V[x] = / a [, )k}} s a mod nterval specal quas dual number polynomal subset strong specal type of topologcal subspace whch s not a mod topologcal deal. Study n ths drecton s nnovatve as k = 1k and s left as an exercse to the reader.

154 MOD Subset Specal Type of nterval 15 Example.5. Let F = {P(G[x])} = {collecton of all subsets from G[x] = { a x / a [, 1) k }} be the mod nterval natural neutrosophc polynomal subset collecton. {F, +}, {F, }, {F, }, {F, } and {F, o } be the mod nterval natural neutrosophc polynomal subset semgroup. Let F o, F, F, F, F, F, F, F, F and F be the mod nterval natural neutrosophc polynomal subset specal type of topologcal spaces assocated wth these semgroups. F has mod topologcal natural neutrosophc zero dvsors and nlpotents and mod topologcal usual zero dvsors. Next we proceed onto brefly descrbe the mod nterval fnte degree polynomal subsets and subset coeffcent polynomals n the followng by examples. Example.55. Let S = P(M[x] 9 ) = {collecton of all subsets from M[x] 9 = { 9 a x fnte degree polynomal subsets collecton. / a [, ), x 1 = 1}} be the mod nterval {S, +}, {S, }, {S, } and {S, } be the mod nterval fnte degree polynomal subset semgroups. S o, S, S, S, S and S be the mod nterval fnte degree polynomal subset specal type of topologcal spaces. Clearly o(s) =. Let A = {.1x +.,.1x +.x + 1.1} and B = {.1x +.,.5x 5 +.x + x +.1,.7 +.1x} S. Clearly A B = {.1x +.}

155 15 MOD Natural Neutrosophc Subset Topologcal A B = {.1x +.,.1x +.x + 1.1,.5x 5 +.x + x +.1,.7 +.1x} A + B = {.x +.,.1x +.1x +.x + 1.,.1x +.5x 5 +.x + x +.,.5x 5 +. x +.x + 1.,.1x +.1x + 1,.1x +.1x A B = {.1x 1 +.x +.9,.1x 8 +.x x +.x +.9x +.,.5x x 5 +.x x +.x 7 +.x +.1 x,., 1.5x x 5 +.x +.1x +.15x +.18 x +.x +.55 x 5 +.x +.x +.x +.11,.7x x 7 +.x,.17 x +. x +.11x} V Clearly the equatons,, and V are dstnct forcng all the mod nterval fnte polynomal specal type of topologcal spaces to be dstnct. Let A = {x +, x 8 + x + } and B = {x +, x + x + } S. Clearly A B = {}. Thus S has mod topologcal zero dvsors but has no mod topologcal nontrval dempotents. Further S has mod nterval fnte degree polynomal subset strong specal type of topologcal subspaces but has no mod nterval topologcal strong deals. All deals are also of nfnte order as o(s) =. Example.5. Let H = {P(W[x] 9 )} = {collecton of all subsets from W[x] = { 9 a x / a [, 1), x 1 = 1}} be the mod nterval natural neutrosophc fnte degree polynomal subsets collecton. {H, +}, {H, }, {H, }, {H, } and {H, } be the mod nterval natural neutrosophc fnte degree polynomal subset semgroup.

156 MOD Subset Specal Type of nterval 155 H o, H, H, H, H, H, H, H, H and H be the mod nterval natural neutrosophc fnte degree polynomal specal type of topologcal spaces assocated H. o(h) =. Example.57. Let D = {P(E[x] 1 )} = {collecton of all subsets from E[x] 1 = { 1 a x / a [, 5) ; x 1 = 1}} be the mod nterval natural neutrosophc-neutrosophc polynomal subset collecton. {D, +}, {D, }, {D, }, {D, } and {D, } be the mod nterval natural neutrosophc-neutrosophc polynomal subset semgroups. D o, D, D, D, D, D, D, D, D and D be the mod nterval natural neutrosophc-neutrosophc polynomal subset specal type of topologcal spaces related wth the 5 mod nterval semgroups. Ths has fnte order mod nterval natural neutrosophcneutrosophc polynomal subset strong topologcal subspaces. Example.58. Let M ={P(F[x] 1 )} = {collecton of all subsets from F[x] 1 = { 1 a x / a C ([, 1)), x 11 = 1}} be the mod nterval natural neutrosophc fnte complex number fnte degree polynomal subset collecton. {M, +}, {M, }, {M, }, {M, } and {M, } be the mod nterval natural neutrosophc fnte complex number fnte degree polynomal semgroup. All the 1 mod nterval natural neutrosophc fnte complex number fnte degree polynomal specal topologcal spaces are dstnct.

157 15 MOD Natural Neutrosophc Subset Topologcal Further ther related propertes can be derved wth approprate modfcatons. Example.59. Let G = P(L[x] ) = {collecton of all subsets from L[x] = { a x / a [, 1) g, x 7 = 1}} be the mod nterval dual number fnte degree polynomal subsets. G o, G, G, G, G and G be the mod nterval dual number fnte degree polynomal subset specal type of topologcal spaces. All propertes can be derved wth approprate modfcatons. n vew of ths we have the followng theorem. Theorem.. Let G = P(S[x] n ) = {Collecton of all subsets from S[x] = { n a x / a [,m) (or [, m) or C([, m)) or C ([,m)), or [, m) or [, m) or [, m) g or [, m) g or [, m) h or [, m) h or [, m) k or [, m) k ); n <, m < }} be the mod nterval fnte degree polynomal subset collecton. ) {G, +}, {G, }, {G, }, {G, } and {G, } be the fve mod nterval fnte degree polynomal subset semgroup where all the operatons are relevant. ) G o, G, G,G, G, G, G, G, G and G are the mod nterval fnte degree polynomal subset specal type of topologcal spaces. ) G has mod nterval fnte degree polynomal subset specal type of strong topologcal subspaces.

158 MOD Subset Specal Type of nterval 157 v) G has mod nterval fnte degree polynomal subset specal type of topologcal deals and not strong deals. v) G has mod nterval fnte degree polynomal subset specal type of topologcal zero dvsors or nlpotents for approprate values of m. v) v) v s true when [, m) g or [, m) g are taken. G has no mod nterval topologcal dempotents. Proof of these can be gven wth approprate modfcatons so left as an exercse to the reader. Example.. Let H = { 9 a x / a P ([, )), x 1 = 1} be the mod nterval subset coeffcent polynomal collecton, H o, H, H, H, H and H be the mod nterval specal topologcal spaces o(h) =. All related propertes can be studed for H wth approprate modfcaton. Example.1. Let K = { ax / a P([, 1) g ), x 1 = 1}, g = } be the mod nterval subset coeffcent polynomal collecton. Let K o, K, K, K, K, K, K, K, K and K be the mod nterval subset coeffcent polynomal of fnte degree specal topologcal space K has mod nterval subset strong topologcal subspaces whch are not topologcal deals. K has nfnte number of zero dvsors and nlpotents. Example.. Let Q = { 5 ax / a P([, 11) k ), x = 1} the mod nterval natural neutrosophc specal quas dual

159 158 MOD Natural Neutrosophc Subset Topologcal number subset coeffcent polynomal collecton. Q o, Q, Q,, Q be the 1 mod nterval topologcal spaces. All related propertes can be derved wth condtons x = 1 and k = 11k. The theorem can be derved for these mod nterval topologcal spaces wth subset coeffcent polynomals of fnte degree wth approprate changes n that theorem. reader. We suggest the followng problems for the nterested PROBLEMS 1. Let W = P([, n)) be the mod nterval subset collecton. {W, +}, {W, }, {W, } and {W, } be the mod nterval subset semgroups. ) Fnd dempotents f any n {W, +} and {W, }. ) ) v) Prove every subset of W s an dempotent n case of {W, } and {W, }. Fnd all zero dvsors n {W, }. Defne the sx mod nterval subset type topologcal spaces usng the four mod semgroups and prove they are dstnct. v) Prove these mod nterval subset topologcal spaces have mod nterval specal type of topologcal zero dvsors, dempotents and nlpotents dependng on n of [, n). v) v) Fnd all mod nterval strong subset topologcal spaces of both fnte and nfnte order. Prove W has no mod nterval specal type of strong subset topologcal deals.

160 MOD Subset Specal Type of nterval 159 v) Obtan any other specal feature assocated wth W.. Let V = {P([, ))} be the mod nterval subset collecton. {V, +}, {V, }, {V, } and {V, } be the mod nterval subset semgroups. Study questons () to (v) of problem (1) for ths V.. Let X = {P([, ))} be the mod nterval subset collecton of [, ). {X, }, {X, +}, {X, }, {X, } be the mod nterval subset semgroups. Study questons () to (v) of problem (1) for ths X.. Let Y = {P([, 5))} be the mod nterval subset collecton of [, 5). ) Study questons () to (v) of problem (1) for ths Y. ) Compare Y wth X of problem () and V of problem (). 5. Let A ={P([, 5) g)} be the mod nterval subset dual number collecton. {A, +}, {A, }, {A, } and {A, } be the mod nterval subset dual number semgroups. ) Study questons () to (v) of problem (1) for ths A. ) Compare A of ths problem wth V of problem ().. Let B = {P([,) g)} be the mod nterval subset dual number collecton. {B, +}, {B, }, {B, } and {B, } be the mod nterval subset dual number semgroups.

161 1 MOD Natural Neutrosophc Subset Topologcal ) Study questons () to (v) of problem (1) for ths B. ) Compare ths B wth problem (5) of A. ) Prove both A and B have nfnte number of zero dvsors. 7. Let C = {P([, ) g)} be the mod nterval subset dual number subsets {C, +}, {C, }, {C, } and {C, } be the mod nterval subset dual number semgroups. ) Study questons () to (v) of problem (1) for ths C. ) Compare ths (C) wth A and B of problems (5) and () respectvely. 8. Let D = {P([, ) )} be the mod nterval neutrosophc subset collecton. {D, }, {D, }, {D, } and {D, +} be the mod nterval neutrosophc subset semgroups. ) Study questons () to (v) of problem (1) for ths D. ) Compare ths D wth C of problem (7). 9. Let E = {P([, ) )} be the mod nterval neutrosophc subset collecton. ) Study questons () to (v) of problem (1) for ths E. ) Compare ths E wth D of problem 8. ) Obtan all the specal features assocated wth ths E. 1. Let F = {P([, ) h)} be the mod nterval specal dual lke number subset collecton. {F, +}, {F, }, {F,

162 MOD Subset Specal Type of nterval 11 } and {F, } be the mod nterval subset specal dual lke number semgroups. ) Study questons () to (v) of problem (1) for ths F. ) Compare ths F wth E and D of problems (9) and (8) respectvely. ) Study all the specal features assocated wth ths F. 11. Let G = {P([, 5) h)} be the mod specal dual lke number nterval subset collecton {G, +}, {G, }, {G, } and {G, } be the mod specal dual lke number nterval subset collecton. ) Study questons () to (v) of problem (1) for ths G. ) Compare ths G wth F of problem (1). 1. Let H = {P(C[, ))} = collecton of all mod nterval fnte complex number subset collecton. {H, +}, {H, }, {H, } and {H, } be the mod subset nterval fnte complex number semgroups. ) Study questons () to (v) of problem (1) for ths H. ) Compare ths H wth G of problem Let J = {P(C[, 59))} be the mod nterval fnte complex number subset collecton. {J, +}, {J, }, {J, } and {J, } be the mod nterval fnte complex subset semgroups. ) Study questons () to (v) of problem (1) for ths J. ) Compare ths J wth H of problem (1).

163 1 MOD Natural Neutrosophc Subset Topologcal 1. Let K = {P([, 8) k)} be the mod nterval specal quas dual number subset collecton. ) Study questons () to (v) of problem (1) for ths K. ) Compare ths K wth J of problem Let L = {P([, 7) k )} be the mod subset nterval specal quas dual number collecton {L, +}, {L, }, {L, } and {L, } be the mod subset nterval specal quas dual number semgroups. ) Study questons () to (v) of problem (1) for ths L. ) Compare ths L wth K of problem 1. ) Enlst all the specal features assocated wth L. 1. Let S = P([, 8)) = collecton of all subsets from mod nterval natural neutrosophc numbers. {S, +}, {S, }, {S, } and {S, } be the mod natural neutrosophc nterval subset semgroups. ) Study questons () to (v) of problem (1) for ths S. ) ) Prove {S, +} has dempotents. Prove {S, } has both nterval zero domnant zero dvsors under and has mod natural neutrosophc zero domnated zero dvsors. 17. Let T = {P( [, 9))} be the mod nterval natural neutrosophc subset collecton. {T, +}, {T, }, {T, } and {T, } be the mod nterval natural neutrosophc subset semgroups. ) Study questons () to (v) of problem (1) for ths T.

164 MOD Subset Specal Type of nterval 1 ) Study questons () and () of problem (1) for ths T. ) Compare ths T wth S of problem (1). v) Enumerate all specal features assocated wth ths T. 18. Let M = {P(C ([, ))} be the mod natural neutrosophc nterval fnte complex number subsets. {M, +}, {M, }, {M, } and {M, } be the mod nterval natural neutrosophc fnte complex number subset semgroup. ) Study questons () to (v) of problem (1) for ths M. ) Study questons () and () of problem (1) for ths M. ) Compare M wth T of problem (17). 19. Let N = {C ([, 1))} be the mod nterval natural neutrosophc fnte complex number subset collecton. {N, }, {N, +}, {N, } and {N, } be the mod nterval natural neutrosophc fnte complex number subset semgroups. ) Study questons () to (v) of problem (1) for ths N. ) Study questons () and () of problem (1) for ths N. ) Compare ths N wth M of problem (18). v) Enumerate all specal features assocated wth ths N.. Let O = {P([, 8) )} be the collecton of all mod nterval natural neutrosophc-neutrosophc subset collecton. {O, }, {O, }, {O, } and {O, +} be the mod nterval natural neutrosophc-neutrosophc subset

165 1 MOD Natural Neutrosophc Subset Topologcal semgroups. O o, O, O, O, O and O be the mod natural neutrosophc-neutrosophc subset specal type of topologcal spaces. ) Study questons () to (v) of problem (1) for ths O. ) Compare ths O wth N of problem 19. ) Compare O wth M of problem Let P = {P([, 17) ) be the mod nterval natural neutrosophc-neutrosophc subset collecton. {P, }, {P, }, {P, } and {P, } be the mod nterval natural neutrosophc-neutrosophc subset semgroup. P o, P P, P, P and P be the mod nterval natural neutrosophc-neutrosophc subset specal type of topologcal spaces. ) Study questons () to (v) of problem (1) for ths P. ) Compare ths P wth O of problem. ) Compare ths P wth M of problem 18.. Let Q ={P([, ) g )} be the mod natural neutrosophc dual number nterval subset collecton {Q, }, {Q, }, {Q, and {Q, +} be the mod natural neutrosophc dual number nterval subset semgroups. Q o, Q, Q, Q, Q and Q be the mod natural neutrosophc dual number nterval subset specal type of topologcal spaces. ) Study questons () to (v) of problem (1) for ths Q. ) Enumerate all specal features assocated wth ths Q.

166 MOD Subset Specal Type of nterval 15. Let R = {P([, 7) g )} be the mod natural neutrosophc nterval dual number collecton of subsets. {R, +}, {R, }, {R, } and {R, } be the mod natural neutrosophc nterval dual number subset. R o, R, R, R, R and R be the mod natural neutrosophc nterval dual number specal type of topologcal spaces ) Study questons () to (v) of problem (1) for ths R. ) Compare ths R wth Q of problem (). ) Obtan all specal features assocated wth R. v) Prove R has nfnte number of zero dvsors and nlpotents.. Let S = {P([, 1) h )} be the collecton of all mod natural neutrosophc specal dual lke number subset collecton. {S, +}, {S, },{S, } and {S, } be the mod nterval natural neutrosophc specal dual lke number subset semgroups. S o, S, S, S, S and S be the mod nterval natural neutrosophc specal dual lke number subset specal type of topologcal spaces. ) Study () to (v) of problem (1) for ths S. ) Enumerate all specal features assocated wth S. v) Compare ths S wth R of problem (). 5. Let T = {P([, 9) h )} be the collecton of all mod nterval natural neutrosophc specal dual lke number subset collecton. {T, +}, {T, }, {T, } and {T, } be the mod nterval natural neutrosophc specal dual lke number subset semgroups. T o, T, T, T, T and T be the mod nterval natural neutrosophc specal

167 1 MOD Natural Neutrosophc Subset Topologcal dual lke number subset topologcal spaces assocated wth these semgroups. ) Study () to (v) of problem (1) for ths T. ) Enumerate all specal features enjoyed by ths T. ) Compare ths T wth S of problem ().. Let V = {P([, 15) k )} be the mod nterval natural neutrosophc specal quas dual number subset collecton. {V, +}, {V, }, {V, } and {V, } be the mod nterval natural neutrosophc specal quas dual number subset semgroups and V o, V, V, V, V and V be specal type of topologcal spaces assocated wth these semgroups. ) Study questons () to (v) of problem (1) for ths V. ) Obtan all specal features assocated wth V. ) Compare ths V wth T of problem (5). 7. Let W = {P([, 17) k be the mod nterval natural neutrosophc specal quas dual number subset collecton {W, +}, {W, }, {W, } and {W, } be the mod subset nterval specal quas dual number semgroups. W o, W, W, W, W and W be the mod subset nterval specal quas dual number nterval specal type of topologcal spaces assocated wth these semgroups. ) Study questons () to (v) of problem (1) for ths W. ) Compare ths W wth V of problem ().

168 MOD Subset Specal Type of nterval 17 a1 a 8. Let M = { a / a P([, 15)); 1 5} be the mod a a 5 nterval subset matrx collecton. ) Prove {M, }, {M, n }, {M, } and {M, +} are mod nterval subset matrx semgroups of nfnte order. ) ) Prove all the mod nterval subset matrx specal type of topologcal spaces usng M o, M, M, M, n M and n are dstnct. n M Are these sx spaces dscrete? v) Can these sx spaces be compact? v) v) v) x) Prove M has mod nterval subset matrx strong specal type of topologcal subspaces? Can M have strong mod nterval subset matrx topologcal deal? Fnd all mod nteral subset topologcal zero dvsors and nlpotents of M. Can M have mod nterval subset topologcal dempotents? x) Menton all the classcal propertes of ths topologcal space satsfed by ths M. a1 a a a a 5 a a 7 a 8 9. Let N = { / a P([, )); 1 a9 a1 a11 a 1 a1 a1 a15 a1 1} be the mod nterval subset entres matrx collecton. {N, +}, {N, }, {N, }, {N, n } and {N, } be the mod

169 18 MOD Natural Neutrosophc Subset Topologcal nterval subset matrx semgroups. N o, N, N, N, N, N, n N and n n N N, N, n be the mod nterval subset matrx specal type of topologcal spaces assocated wth these 5 semgroups. ) Study questons () to (x) of problem (8) for ths N. ) Prove N, N, N and N n are mod nterval non-commutatve subset matrx topologcal spaces. ) v) Prove these four spaces have mod nterval topologcal rght zero dvsors whch are not mod nterval topologcal left zero dvsors and vce versa. Prove all these four spaces has mod nterval topologcal subset rght deals whch are not left deals and vce versa. a1 a a a a5. Let V = { a a 7 a8 a9 a / a {P( [, 8))}; 1 1 1} be the mod natural neutrosophc subset nterval matrx collecton. {V, +}, {V, }, {V, and {V, } be the mod natural neutrosophc nterval subset matrx semgroups. V o, V, V, n V, n V and n V be the mod natural neutrosophc subset nterval matrx specal type of topologcal spaces assocated wth these semgroups. ) Study questons () to (x) of problems (8) for ths V. ) Obtan all specal feature assocated wth ths V.

170 MOD Subset Specal Type of nterval 19 a1 a a 1. Let W = { a a5 a / a P( [, )); 1 9} be a7 a8 a 9 the mod natural neutrosophc subset nterval matrx collecton. {W, +}, {W, n }, {W, }, {W, } and {W, } be the mod subset natural neutrosophc nterval semgroups. ) Study questons () to (x) of problem (8) for ths W. ) Study questons () to (v) of problem (9) for ths W. a1 a a a. Let Y = { / a P([, 7) g ); 1 8} be a5 a a 7 a8 the mod nterval natural neutrosophc subset dual number matrces. {Y, }, {Y, }, {Y, +} and {Y, n } be the mod natural neutrosophc nterval dual number subset matrx semgroups and Y o, Y, Y, Y, Y and Y be the mod natural neutrosophc dual number nterval subset matrx specal type of topologcal spaces. ) Study questons () to (x) of problem (8) for ths Y. ) Prove ths Y has more number of zero dvsors and nlpotents. a1 a a a a5 a. Let Z = { a 7 a8 a9 a1 a11 a / a P([, 1 8) g ); 1 1} be the mod natural neutrosophc dual number nterval subset matrx collecton. {Z, +}, {Z, n }, {Z, } and {Z, } be the mod natural n

171 17 MOD Natural Neutrosophc Subset Topologcal neutrosophc subset matrx nterval dual number semgroups and Z o, Z, Z, Z, n Z and n be the n mod natural neutrosophc dual number subset nterval matrx specal type of topologcal spaces. ) Study questons () to (x) of problem (8) for ths Z. Z ) Compare ths Z wth Y of problem (). a1 a a. Let A = { / a P([, ) ), 1 } be the a a 5 a mod natural neutrosophc-neutrosophc nterval subset matrx collecton {A, +}, {A, }, {A, } and {A, n } be the mod nterval natural neutrosophc-neutrospohc subset matrx semgroups. ) Study questons () to (x) of problem (8) for ths A. a1 a a a a 5 a a 7 a 8 5. Let B = { / a P([, 7) ); a9 a1 a11 a 1 a1 a1 a15 a1 1 1} to be mod natural neutrosophc nterval subset matrx collecton {B, +}, {B, }, {B, n }, {B, } and {B, } be the mod natural neutrosophc nterval subset matrx. B o, B, B, B, B, B, B, n B, n B and n B be the mod natural neutrosophc neutrosophc subset nterval matrx specal type of topologcal spaces bult usng these mod semgroups. n

172 MOD Subset Specal Type of nterval 171 ) Study questons () to (x) of problem (8) for ths B. ) Compare B wth A of problem (). ) Obtan all specal features enjoyed by B.. Let C = a a a a a a a a a a a a / a P(C([, ))); 1 1} be the mod nterval fnte complex number subset matrx collecton. {C, +}, {C, n }, {C, } and {C, be the mod nterval fnte complex number subset matrx semgroups. C o, C, C, C, n C and n C be the mod nterval fnte complex number subset matrx specal type of topologcal spaces. ) Study questons () to (x) of problem (8) for ths C. ) Enumerate all specal features enjoyed by ths C. a1 a a 7. Let D = { a a5 a / a P(C([, ))); 1 9} a7 a8 a 9 be the mod nterval fnte complex number subset matrx collecton. {D, +}, {D, }, {D, }, {D, n } and {D, } be the mod nterval fnte complex number subset matrx semgroups. D o, D, D, D, D, D, D, n D, n D and n D be the mod nterval fnte complex number subset specal type topologcal spaces assocated wth the mod semgroups. ) Study questons () to (x) of problem (8) for ths D. ) Compare ths D wth (C) of problem (5). n n

173 17 MOD Natural Neutrosophc Subset Topologcal a1 a a 8. Let F = / a P([, )k); 1 } be the a a 5 a mod nterval specal quas dual number subset matrx collecton. {F, +}, {F, }, {F, } and {F, } be the mod nterval subset specal quas dual number matrx semgroup. F o, F, F, n F, n F and n F be the mod nterval specal quas dual number subset matrx specal topologcal space. ) Study questons () to (x) of problem (8) for ths F. ) Enumerate all specal feature enjoyed by F. a1 a a a a5 9. Let G = { / a P([,1) g), a a 7 a8 a9 a1 1 1} be the mod nterval dual number subset matrx collecton. Study questons () to (x) of problem (8) for ths G. a1 a a a. Let H = { / a P([, 1) h ); 1 8} a5 a a 7 a8 be the mod natural neutrosophc specal dual lke number nterval subset matrx collecton. {H, +}, {H, n }, {H, } and {H, } be the mod natural neutrosophc dual number specal dual lke number nterval matrx subset semgroups.

174 MOD Subset Specal Type of nterval 17 ) Study questons () to (x) of problem (8) for ths H. ) Enumerate all specal features assocated wth ths H. a1 a a a a5 1. Let J = { a a7 a8 a9 a 1 / a P([, 18) a11 a1 a1 a1 a 15 k ); 1 15} be the mod natural neutrosophc nterval specal quas dual number subset matrx collecton. {J, +}, {J, }, {J, } and {J, n } be the mod natural neutrosophc nterval specal quas dual number subset matrx semgroups. J o, J, J, J, n J and n be the mod natural neutrosophc specal quas dual number nter subset matrx specal type of topologcal spaces. ) Study questons () to (x) of problem (8) for ths J. ) Obtan all specal features enjoyed by J. a1 a a a a5 a a 7 a8 a9 a 1. Let K = { / a P(C [, 8)); a11 a1 a1 a1 a 15 a1 a17 a18 a19 a 1 } be the mod natural neutrosophc fnte complex number nterval subset matrx collectons. ) Study questons () to (x) of problem (8) for ths K. ) Fnd all specal features enjoyed by ths K. n J

175 17 MOD Natural Neutrosophc Subset Topologcal. Let M = P(R) = {collecton of a matrx subset from M a1 a where R = { a / a [, 17); 1 5}} be the mod a a 5 nterval matrx subset collecton. {M, +}, {M, n }, {M, } and {M, } be the mod nterval matrx subset semgroups. M o, M, M, M, n M and n be the n M mod nterval matrx subset specal type of topologcal spaces assocated wth these four mod semgroups. ) Prove the four mod semgroups are dstnct and are of nfnte order. ) ) v) Prove the mod matrx subset topologcal spaces are of nfnte order. Are these spaces dscrete? Are these sx mod nterval subset matrx spaces connected? v) Are these spaces compact? v) v) v) x) Can these mod nterval matrx subset spaces have topologcal zero dvsors? Can these mod nterval matrx subset specal type of topologcal spaces have topologcal nlpotents? Prove some of these mod nterval subset specal type of topologcal spaces have topologcal dempotents. Prove M has strong mod nterval matrx subset specal type of topologcal subspaces. x) Prove M has no strong mod nterval matrx subset topologcal deals. x) Prove all deals of M are of nfnte order.

176 MOD Subset Specal Type of nterval 175 x) Prove M has mod nterval matrx subset topologcal subspaces of fnte order.. Let T = P(B) = {collecton of a matrx subset from a1 a B = { a a where a [, ); 1 } be mod nterval neutrosophc nterval matrx subset collecton. Let {T, +}, {T, n }, {T, }, {T, } and {T, } be the mod natural neutrosophc nterval matrx subset semgroup. T o, T, T, T, T T, n T, T, n T and n n T be the mod natural neutrosophc nterval matrx subset specal type of topologcal spaces assocated wth the 5 mod semgroups. ) Prove T, T T and n T are non commutatve natural neutrosophc nterval matrx subset specal type of topologcal spaces. ) Study questons () to (x) of problem () for ths T. ) Compare ths T wth M of problem (). v) Obtan all specal features assocated wth T. 5. Let V = {P(L) = {collecton of all matrx subsets from a1 a a a a5 L = { a a 7 a8 a9 a / a C([, )); 1 1 1}} be mod nterval fnte complex number matrx subset collecton. {V, n }, {V, }, {V, } and {V, +} be the mod nterval fnte complex number matrx subset semgroups. V o, V, V, n V, n V and n V be the mod nterval fnte complex number matrx subset specal type of topologcal spaces related wth the four mod semgroups. ) Study questons () to (x) of problem () for ths V.

177 17 MOD Natural Neutrosophc Subset Topologcal ) Study questons (), () and (v) of problem () for ths V.. Let W ={P(B) = collecton of all matrx subsets from a1 a a B = { a a5 a / a [, 1) g; 1 9}} be a7 a8 a 9 the mod nterval dual number matrx subset collecton. Let {W, +}, {W, n }, {W, }, {W, }, {W, } be the mod nterval dual number matrx subset semgroups. W o, W, W, W, W, W, n W, n W and W be n the mod nterval dual number matrx subset specal type of topologcal spaces. ) Study questons () to (x) of problem () for ths W. ) Compare ths W wth V of problem 5. ) Derve all the specal features assocated wth ths W. 7. Let Y = {P(D) = collecton of all subsets from a1 a D = { a / a [, ) ; = ; 1 5}} be the a a 5 mod nterval neutrosophc matrx subset collecton. {Y, +}, {Y, }, {Y, } and {Y, n } be the mod nterval neutrosophc matrx subset semgroup. Y o, Y Y, Y n, Y and n Y be the mod nterval neutrosophc matrx subset specal type of topologcal spaces usng the four mod semgroups. ) Study questons () to (x) of problem () for ths Y. n

178 MOD Subset Specal Type of nterval 177 ) Derve all the specal features enjoyed by Y. ) Compare ths Y wth W of problem (). 8. Let Z = {P(E) where a1 a a a a5 a E ={ a7 a8 a9 a1 a11 a1 a1 a1 a15 a1 a17 a 18 where a [, ) k); 1 18} be the mod nterval specal quas dual number matrx subsets collecton. {Z, +}, {Z, }, {S, } and {Z, n } be the mod nterval specal quas dual number matrx subset semgroups. Z o, Z, Z, Z, Z and Z be the mod nterval specal quas dual number matrx subset specal type of topologcal spaces assocated wth the semgroup. Study questons () to (x) of problem () for ths Z. 9. Let P = {P(W) collecton of all subsets from a1 a W = { a / a [, ); 1 5}} be the mod a a 5 nterval natural neutrosophc matrx subset collecton. ) Study questons () to (x) of problem () for ths P. ) Compare ths P wth Z of problem (8). ) Derve all specal propertes assocated wth P.

179 178 MOD Natural Neutrosophc Subset Topologcal 5. Let B = {P(S), collecton of all matrx subsets from a1 a a a a5 S = { a a7 a8 a9 a 1 / a C ([, )); a11 a1 a1 a1 a }} be the mod natural neutrosophc nterval fnte complex number matrx subset collecton. {B, +}, {B, n }, {B, } and {B, } be the mod natural neutrosophc nterval fnte complex number matrx subset semgroup. B o, B, B, B, n B and n B be n the mod natural neutrosophc nterval fnte complex number matrx subset specal type of topologcal spaces. ) Study questons () to (x) of problem () for ths B. ) Compare ths B wth P of problem (9). 51. Let D = {P(L), collecton of matrx subsets from a1 a a a a 5 a L = { / a {[, ) }; 1 1}} be a7 a8 a9 a 1 a11 a1 the mod nterval natural neutrosophc neutrosophc matrx subset collecton. Let {D, +},{D, n }, {D, } and {D, } mod nterval natural neutrosophcneutrosophc matrx subset semgroups. D o, D, D, n D, n D and n D be the mod nterval natural neutrosophc-neutrosophc matrx subset topologcal spaces relatve wth the four mod semgroups. ) Study questons () to (x) of problem () for ths D.

180 MOD Subset Specal Type of nterval 179 ) Compare ths D wth B of problem Let E = {P(M) = {collecton of all subsets from a1 a M = { a / a {15 g }; 1 1} be the mod a 1 nterval natural neutrosophc dual number matrx subset collecton. {M, +}, {M, n }, {M, }and {M, } be the mod nterval natural neutrosophc dual number matrx subset semgroup. M o, M, M, n M, n M and n M be the mod nterval natural neutrosophc dual number matrx subset specal type of topologcal spaces. ) Study questons () to (x) of problem () for E. ) Compare ths E wth D of problem F = {P(S) = {collecton of all matrx subsets from a1 a a S = { a a5 a / a [, ) h ; 1 9}} be a7 a8 a 9 the mod nterval natural neutrosophc specal dual lke number matrx subsets collecton. {F, +}, {F, n }, {F, }, {F, } and {F, } be the mod nterval natural neutrosophc specal dual lke number matrx subset semgroups.

181 18 MOD Natural Neutrosophc Subset Topologcal F o, F, F, F, n F, F, F, F, n F and n n F be the mod nterval natural neutrosophc specal dual number matrx subset specal types of topologcal spaces. ) Study questons () to (x) of problem () for ths F. ) Compare ths F wth E of problem (5). a1 a a a a 5 5. Let G = {P(M), where M = { / a a7 a8 a9 a1 a [, 1) k ; 1 1}} be the mod nterval natural neutrosophc specal quas dual number matrx subset collecton. {G, +}, {G, n }, {G, } and {G, } be the mod nterval natural neutrosophc specal quas dual number matrx subset semgroup, G o, G, G, G, n G and n G be the mod nterval natural neutrosophc quas dual number matrx subset specal type of topologcal spaces. ) Study questons () to (x) of problem () for ths G. ) Compare ths G wth F of problem (5). 55. Let H = {P(L[x]} = {collecton of all subsets from L[x] = { a x n / a [, )}} be the mod nterval polynomal subset collecton. ) o(h) =. ) ) Prove {H, }, {H, }, {H, +} and {H, } be the mod nterval polynomal subsetsemgroup. Prove only {H, }, {H, } and {H, +} can have fnte order mod nterval subsemgroups.

182 MOD Subset Specal Type of nterval 181 v) Prove {H, } cannot have mod subsemgroups of fnte order. v) Prove {H, } and {H, } can have mod nterval dempotents. v) Prove H o, H, H, H, H and H are dstnct mod topologcal spaces. v) v) x) How many mod nterval strong topologcal subspaces of H are there? Prove H has no mod nterval strong topologcal deals. Fnd all mod nterval topologcal zero dvsors of H. x) Does H contan mod nterval topologcal nlpotents? x) Obtan any other specal feature enjoyed by H. 5. Let ={P(T[x])} ={collecton of all subsets from T[x] = { ax / a polynomal subset collecton. [, 5)}} be the mod nterval Study questons () to (x) of problem (55) for ths. 57. Let J = {P(S[x])} = {collecton of all subsets from S[x] = { ax / a C([, 1), F = 11}} be the mod nterval fnte complex number coeffcent polynomal subset collecton. Study questons () to (x) of problem (55) for ths J. 58. Let K = {P(R[x])} = {collecton of all subsets from R[x] = { a x / a C ([, 9))} be the mod nterval

183 18 MOD Natural Neutrosophc Subset Topologcal natural neutrosophc fnte complex number coeffcent polynomal subset collecton. Study questons () to (x) of problem (55) for ths K. 59. Let L = {P(V[x])} = {collecton of all subsets from V[x] = { a x / a ([, 8), = }} be the mod nterval neutrosophc coeffcent polynomal subset collecton. Study questons () to (x) of problem (55) for ths L.. Let N = {P(W[x])} = {collecton of all subsets from W[x] = { a x / a [, 17) }} be the mod nterval natural neutrosophc-neutrosophc coeffcent polynomal subsets collecton. Study questons () to (x) of problem (55) for ths N. 1. Let R = {P(Z[x])} = {collecton of all subsets from Z[x] = { a x / a [, 1) g, g = }} be the mod nterval dual number coeffcent polynomal subset collecton. Study questons () to (x) of problem (55) for ths R.. Let S = {P(Y[x])} = {collecton of all subsets from Y[x] = { a x / a [, 1) g, g = }}be the mod nterval natural neutrosophc dual number coeffcent polynomal subsets collecton. ) Study questons () to (x) of problem (55) for ths S.

184 MOD Subset Specal Type of nterval 18 ) Prove S has more mod nterval zero dvsors and nlpotents. ) Compare ths S wth R of problem 1.. Let V = {P(T[x])} = {collecton of all subsets fron T[x] = { a x / a [, 7) h, h = h}} be the mod nterval specal dual lke number coeffcent polynomal subset collecton. ) Study questons () to (x) of problem (55) for ths V. ) Compare ths V wth S and R of problems () and (1) respectvely.. Let W = {P(B[x])} = {collecton of all subsets from B[x] = { a x / a ( [, 1) h, h = h}} be the mod nterval natural neutrosophc specal dual lke number coeffcent polynomal collecton. ) Study questons () to (x) of problem (55) for ths W. ) Compare ths W wth V of problem (). 5. Let Z = {P(W[x])} = {collecton of all subsets from W[x] = { a x / a [, 18) k, k = 17k}} be the mod nterval specal quas dual number coeffcent polynomal subsets collecton. ) Study questons () to (x) of problem (55) for ths Z. ) Compare ths Z wth W of problem ().

185 18 MOD Natural Neutrosophc Subset Topologcal. Let D = {P(E[x])} = {collecton of all subsets from E[x] = { a x / a [, 5) k, k = k}} be the mod nterval natural neutrosophc specal quas dual number coeffcent polynomal subset collecton. ) Study questons () to (x) of problem (55) for ths D. 7. Let M = {P(F[x] 1 )) = {collecton of all subsets from F[x] 1 = { 1 a x / a [, 1), x 11 = 1}} be the mod nterval coeffcent polynomal subsets collecton. ) o(m) =. ) Prove {M, +}, {M, }, {M, } and {M, } are mod nternal semgroups of nfnte order whch contans subsemgroups of fnte order. ) Prove M o, M, ( M ), M, M, M, M, ( M, M, ) are mod nterval specal type v) M of polynomal subset topologcal spaces. Prove M has mod nterval strong topologcal subspaces but no strong topologcal deals. v) Prove all topologcal deals are of nfnte order. v) v) v) x) Prove for approprate values of n n [, n), M wll have mod nterval topologcal nlpotents and zero dvsors. Prove M has no nontrval mod nterval topologcal dempotents. Prove M has mod nterval strong topologcal subspaces of fnte order. Obtan any other specal feature assocated wth M.

186 MOD Subset Specal Type of nterval Let N = {P(B[x] 1 )} = {collecton of all subsets from B[x] 1 = { a x / a [, ), x 17 = 1}} be the mod nterval natural neutrosophc coeffcent fnte degree polynomals subset collecton. Study questons () to (x) of problem (7) for ths N. 9. Let P = {P(D[x] 7 )} = {collecton of all subsetsfrom D[x] 7 = { 7 a x / a [, 1) g; g =, x 8 = 1}} be the mod nterval dual number coeffcent of fnte degree polynomal collecton. Study questons () to (x) of problem 7 for ths P. 7. Let F = {P(G[x] )} = {collecton of all subsets from G[x] = { a x / x 7 = 1 a [, 1) g }} be the mod nterval natural neutrosophc dual number coeffcents polynomals of fnte degree subsets collecton. Study questons () to (x) of problem (7) for ths F. 71. Let H = {P(L[x] )} = {collecton of all subsets from L[x] = { a x / a C ( [, 1)), = 11, F x = 1}} be the mod nterval natural neutrosophc fnte complex number coeffcent polynomals of fnte degree subsets collecton. Study questons () to (x) of problem (7) for ths H. 7. Let E = {P(T[x] 8 )} = {collecton of all subsets from T[x] 8 = { 8 a x / a [, 1), x 9 = 1, = }} be the mod nterval natural neutrosophc neutrosophc

187 18 MOD Natural Neutrosophc Subset Topologcal coeffcent polynomals of fnte degree subset collecton. Study questons () to (x) of problem (7) for ths E. 7. Let M = {P(T[x] 9 )} ={collecton of all subsets from T[x] 9 = { 9 a x / a [, 1) k, x 1 = 1, k = 11k}} be the mod nterval natural neutrosophc specal quas dual number coeffcent polynomal of fnte degree subsets. ) Study questons () to (x) of problem (7) for ths M. ) Compare ths M wth E and H of problems (7) and (7) respectvely. 7. Let T = { a x / a P ([, 17))} be the mod nterval subset coeffcent polynomal collecton ) o(t) =. ) ) v) Prove {T, +}, {T, }, {T, } and {T, } are mod nterval subset coeffcent polynomal semgroups. n {T, } and {T, } every element s an mod dempotent. Fnd mod subsemgroups and deals of these four mod semgroups. v) Prove T o, T, T, T, T and T are mod nterval subset coeffcent polynomal specal type of topologcal spaces. v) v) Fnd all mod nterval subset coeffcent polynomal strong topologcal subspaces. Fnd all mod nterval subset coeffcent topologcal zero dvsors and nlpotents.

188 MOD Subset Specal Type of nterval 187 v) x) Prove T has no nontrval mod nterval subset coeffcent topologcal dempotents n T, T and T. Prove T has no mod nterval strong topologcal deals. 75. Let S = { a x / a P( [, ))} be the mod nterval natural neutrosophc subset coeffcent polynomal collecton. ) Study questons () to (x) of problem (7) for ths T. ) ) v) Prove {S, o } and {S, } are two dstnct mod nterval natural neutrosophc subset coeffcent polynomal semgroups. T, T, T and T are also dfferent and dstnct collecton of mod nterval natural neutrosophc subset coeffcent polynomal topologcal spaces. Show n general a mod nterval topologcal zero dvsor s not a mod nterval topologcal natural neutrosophc zero dvsor and vce versa. v) Prove S can have mxed mod nterval topologcal zero dvsors. 7. Let R = { a x / a P(C[, 1))} be the mod nterval fnte complex number subset coeffcent polynomal collecton. Study questons () to (x) of problem (7) for ths R.

189 188 MOD Natural Neutrosophc Subset Topologcal 77. Let W = { a x / a C ([, 1))} be the mod nterval natural neutrosophc fnte complex number subset coeffcent polynomal collecton. ) Study questons () to (x) of problem (7) for ths W. ) 78. Let V = { Study questons () to (v) of problem (75) for ths W. a x / a P([, ) )} be the mod nterval neutrosophc subset coeffcent polynomal collecton. Study questons () to (x) of problem (7) for ths V. 79. Let D = { a x / a P([, 1) )} be the mod nterval natural neutrosophc subset coeffcent polynomal collecton. ) Study questons () to (x) of problem (7) for ths D. ) 8. Let E = { Study questons () to (v) of problem (75) for ths D. a x / a P([, 1) g)} be the mod nterval dual number subset coeffcent polynomal collecton. ) Study questons () to (x) of problem (7) for ths E. ) Ths E has more number of mod nterval topologcal zero dvsors and nlpotents.

190 MOD Subset Specal Type of nterval Let H = { a x / a P([, ) g ) be the mod nterval natural neutrosophc dual number subset coeffcent polynomal collecton. ) Study questons () to (x) of problem (7) for ths H. ) ) 8. Let G = { Study questons () to (v) of problem (75) for ths H. Prove ths H has more number of mod nterval topologcal zero dvsors, nlpotents, natural neutrosophc zero dvsors and nlpotents and mxed zero dvsors and nlpotents. a x / a P([, ) h )} be the mod nterval natural neutrosophc specal dual lke number subset coeffcent collecton. ) Study questons () to (x) of problem (7) for ths G. ) Study questons () to (v) of problem (75) for ths G. ) Compare ths G wth H of problem Let K = { a x / a P([, 1) k), k = 1k}} be the mod nterval specal quas dual number subset coeffcent polynomal collecton. Study questons () to (x) of problem (7) for ths K. 8. Let L ={ 1 a x / a P([, 1)), x 1 = 1} be the mod nterval subset coeffcent polynomal collecton.

191 19 MOD Natural Neutrosophc Subset Topologcal ) Study questons () to (x) of problem (7) for ths L. ) Derve all the specal features enjoyed by L. 85. Let M ={ a x / a P( [, 17), x 7 = 1} be the mod nterval natural neutrosophc subset coeffcent fnte degree polynomal collecton. ) Study questons () to (x) of problem (7) for ths M. ) Study questons () to (v) of problem (85) for ths M. ) Compare ths M wth L of problem Let R = { a x / a P(C ([, 5))), x 7 = 1} be the mod nterval natural neutrosophc fnte complex number subset coeffcent polynomal collecton. ) Study questons () to (x) of problem (7) for ths R. ) Study questons () to (v) of problem 85 for ths R. ) Compare ths R wth M of problem 85.

192 Chapter Four MOD SUBSET TOPOLOGCAL SPACES ON THE MOD PLANE R n (m) AND KAKUTAN S THEOREM The study of modulo values when we restrct to the frst quadrant of the real plane R n (m) = [, m) [, m) ={(a, b) / a, b [, m)} s defned as the MOD plane []. Algebrac operatons are performed on R n (m) n []. The plane s represented by the followng fgure. m t s a half open square. m Fgure.1

193 19 MOD Natural Neutrosophc Subset Topologcal Let P(R n (m)) = {collecton of all subsets from R n (m)}, we defne the operatons and on P(R n (m)). P(R n (m)) s an nfnte collecton wth R n (m) as the greatest element under and as the least element under. nfact R n (m) s an nfnte dstrbutve lattce. Thus {R n (m),, } can be realzed as a usual topologcal space of nfnte order. One can also vsualze R n (m) as a truncated topologcal space of the real plane. Here we call R n (m) as the MOD plane subset specal type of topologcal space. The advantage of buldng ths type of MOD plane subset specal type of topologcal space s that we can get nfnte number of them n contrast wth only one such real plane topologcal space. The propertes of or happens to result n a semlattce on P(R n (m)). So as per need one can use these MOD plane subset topologcal spaces. We wll llustrate ths stuaton by some examples. Example.1. Let R n (5) = {(a, b) / a, b [, 5)} be the MOD plane S = P(R n (5)) = {collecton of all subsets of R n (5)}. {S,, } s a MOD topologcal space of nfnte order. nfact {S,, } {P(R R),, } s a subspace of {P(RR),, }. Several open problems can be proposed at ths stage for any R n (m); m <. Lkewse {(, ),, } s a topologcal space so {[, m),, } s a subspace, here when we call or defne MOD nterval we assume n that ntervals when some operaton lke + s performed m = (mod n) so we cannot say m 1 + 1

194 MOD Subset Topologcal Spaces on the 19 s undefned n our spaces t takes the value gven by m (mod m). So we have to work n a dfferent drecton the Kakutan s fxed pont theorem whch s a challengng problem assocated wth MOD subset nterval topologcal spaces. We wll llustrate ths stuaton by some more examples. Example.. Let S ={P(R n (1)) = {collecton of all subsets from R n (1) = {(a, b) / (a, b) [, 1) [, 1)} be the collecton of MOD plane subsets. {S,, } s the MOD plane subset specal type of topologcal space. Let A = {(,.5), (.7,.11), (11.5, 9.75), (1.1,.5), (.11,.1), (,.7), (5., 1)} and B = {(,.5), (1.1,.5), (5., 1), (,.55), (8., 7), (1, 1.5), (.887,.1)} S. Clearly A B ={(,.5), (.7,.11), (11.5, 9.75), (1.1,.5), (.11,.1), (,.7), (5., 1), (,.55), (8., 7), (1, 1.5), (.887,.1)} A B ={ (,.5), (1.1,.5), (5., 1)} Ths s the way operatons are performed on S and (S,, } s defned as a MOD subset plane topologcal space. The followng observatons are pertnent, we have already n the earler chapters defned MOD nterval specal type of topologcal spaces usng the dual plane [, m) g, the neutrosophc plane [, m), the fnte complex modulo nteger plane C([, m)), the MOD nterval specal quas dual number plane [, m) k and MOD nterval specal dual lke number plane [, m) h.

195 19 MOD Natural Neutrosophc Subset Topologcal We could not and have not done such study about MOD nterval real plane [, m) [, m) = R n (m). Ths s study s manly taken here to fnd the valdty of Kakutan s theorem. For these planes are new so such study may not be exstng n case of Rg ={a + bg / a, b R, R reals} and so on. Only for C, the complex plane such study s complete n lterature. Now lkewse the MOD nterval specal type of topologcal space bult usng P([, m)); m < s thoroughly studed n chapters and of ths book. Now the nterval [, 1) does not satsfy the Kakutan s theorem for the space s not complete but we n our very defnton of MOD nterval [, m), assume m (mod m). Hence we propose the followng conjectures. Conjecture.1. Let P([, m)) = T be the MOD nterval subset collecton. {T,, } s a MOD nterval subset specal type of topologcal space n whch m (mod m), can under ths stuaton T satsfy the Kakutan s theorem? Conjecture.. Study conjecture.1 by replacng [, m) by each one of the followng [, m) g, [, m) h, [,m) k, C([ m)) and [, m). All these are MOD ntervals or MOD planes closed under the bnary operatons +,, and. Conjecture.. Let {S, +, } where S = P([, m)) be the MOD nterval specal type of subset topologcal space. Can {S, +, } satsfy the Kakutan s theorem? Conjecture.. n conjecture. replace [, m) by each one of them [, m) g, [, m) h, C([, m)), [, m) and [,

196 MOD Subset Topologcal Spaces on the 195 m) k and analyse the valdty or otherwse of the Kakutan s conjecture. Conjecture.5. Let {P([, m)), +, }, {P([, m), +, }, {P([, m),, and {P([, m)),, } be the four dstnct MOD nterval specal type of subset topologcal spaces. Can these four spaces satsfy Kakutan s theorem? Conjecture.. Can Kakutan s theorem be true for the followng MOD nterval subset specal type of topologcal spaces: ) {P([, m) g), + }, {P([, m), +, }, {P([,m) h), +, }, {P([,m) k), +, } and {P(C[, m)), +, }? ) Can Kakutan s theorem be true n case of the followng MOD nterval subset specal type of topologcal spaces. {P([, m) g, +, }, {P([, m) h), +, }, {P ([, m) k), +, }, {P([, m), +, } and {P(C([, m)), +, }? (Study f {+, } s replaced by {, } or {, } on the above fve spaces). Ths type of study s new and nnovatve so only we have proposed these open conjectures and they happen to be dffcult. Can we fnd fxed ponts as gven by the Kakutan s theorem? Clearly they are part of a plane or lne. The plane or lne s deal and satsfes Kakutan s theorem. We are not sure whether these propertes are nherted by the MOD ntervals and MOD planes.

197 19 MOD Natural Neutrosophc Subset Topologcal Now we develop MOD plane subset specal type of topologcal spaces usng, +,, and operatons. To ths end we gve examples of MOD plane subset semgroups. Example.. Let S = P(R n ()) = {P([, ) [, )) = {collecton of all subsets from R n () = [, ) [, )}} be the MOD plane subsets collectons. {S, } s a semgroup, n fact a semlattce. Smlarly {S, } s a semgroup whch s a semlattce. {S, +} s a MOD subset plane semgroup. {S, } s a MOD subset plane semgroup. Let A = {(., 5), (.1,.), (.1,.5), (.1, ), (1,.5)} and B = {(.1, ), (.,.5), (.1,.5), (.5, ), (,.), (.1,1.5)} S = P(R n ()). We fnd A B ={(., ), (.1,.), (.1, ), (.1, ), (.1,.1), (.,.5), (.,.1), (.,.5), (., ), (.,.5), (.1,.5), (.1,.1), (.1,.5), (.1, ), (.1,.5), (.15, ), (.5, ), (., ), (.5, ), (.5, ), (, 1.5), (,.), (, 1.5), (,), (,.15), (., 5.5), (.1,.1), (.1,.75), (.1, ), (.1,.55)} A + B = {(.1, 1), (.11,.), (.1,.5), (.11, ), (1.1,.5), (., 5.5), (.1,.7), (.1,.55), (.1,.5), (1.,.55), (.1, 5.5), (.11,.5), (.11,.55), (.,.5), (1.5,.5), (.5, 5), (.15,.5), (.15,.), (., ), (1.5,.5), (., 5.), (.1,.), (.1,.8), (.1,.), (1,.5), (.,.5), (., 1.7), (., 5.55), (.11, 1.5), (1.1, 1.55)} Ths s the way + and operatons are performed on P(R n ()).

198 MOD Subset Topologcal Spaces on the 197 We call these (S, +} and {S, } as MOD plane semgroups. Hence we defne {S, +, }, {S,, }, {S,, }, {S, +, } and {S, +, } for S = {P(R n (m))}; m < as the MOD plane subset specal type of topologcal spaces. We see all these sx MOD plane subset topologcal spaces are dstnct and enjoy dfferent types of propertes. We wll llustrate these stuatons by some examples. Example.. Let B = P(R n (7)) = {collecton of all subsets from the MOD plane R n (7) ={(a, b) / a, b [, 7)}} be the MOD plane subsets collecton. {B, +}, {B, }, {B, } and {B, } are the four MOD plane subset semgroups. B o = {B,, }, B = {B, +, }, B = {B, B B and B = {B,, } are the MOD plane subset specal type of topologcal spaces assocated wth the four MOD plane subset semgroup. Can we get Kakutan s theorem to be true n case of these MOD plane subset specal type of topologcal spaces? For these are just subset of the real plane, so B o can nfact be realzed as the specal type of proper subspace of ths MOD plane subset topologcal space. Study n ths drecton s not only nnovatve and nterestng, these results are left as open conjectures. That s valdty of the Kakutan s conjectures n case of MOD plane subset specal type of topologcal spaces s a challengng one.

199 198 MOD Natural Neutrosophc Subset Topologcal Next we proceed onto descrbe MOD plane subset matrx specal type of topologcal spaces and MOD plane matrx subset specal type of topologcal spaces. Here t s pertnent to keep on record that we can have dfferent types of MOD plane specal topologcal spaces. We wll llustrate ths stuaton by examples. Further authors wsh to state by descrbng the operatons by examples t makes easy for any researcher to understand the concept and that s vtal. Example.5. Let M = {(a 1, a, a, a ) / a P([, 1); 1 } be the MOD nterval subset matrx collecton. We wll ndcate how the operatons are performed on M. Let x = ({., 1.1, }, {1.1,.5, 5}, {.,.}, {,., 1,.}) and y = ({,,,.1}, {.7, 5}, {,.}, {1,.5,.7}) M. x y = ({,, 1.1,,.1,.}, {.5, 5,.7, 1.1}, {,.,.}, {,., 1,.,.5,.7}) x y = ({}, {5}, {.}, {1}) x + y = ({., 1.1,,., 1.1, 7,.,.1, 8,., 1.,.1}, {1.8, 1., 5.7,.1, 5.5, 1}, {.,., 1.,.8}, {5,.5,.7, 1.,, 1., 1.7,.8, 1, 1.5,.7,.9}) x y = ({,.9,., 1.,.,,., 1.1,.}, {(.77,.5,.5, 5.5,.5, 1}, {.,.8,.,.1}, {,., 1,.,,.15,.5, 1,.8,.1,.7,.1}) V All the four equatons are dstnct so (M, +), {(M, }, {M, } and {M, } are four dstnct MOD nterval subset entres matrx semgroups of nfnte order.

200 MOD Subset Topologcal Spaces on the 199 Thus M o ={M,, }, M = {M, +, }, M = {M, +, }, M = {M, +, }, M = {M,, } and M = {M,, are all sx dstnct MOD plane subset matrx entres specal type of topologcal spaces. Ths has MOD nterval subset matrx specal type of topologcal subspaces. nfact all these have MOD nterval subset matrx specal type of topologcal subspaces that are somorphc wth MOD plane subset specal type of topologcal spaces lke P 1 = {({a 1 }, {},{}, {}) /{a 1 } P[, 1))} the MOD plane subset topologcal spaces got from P[, 1) P = {({}, {a }, {}, {}) such that {a } P ([, 1)} P = {({},{}, {a }, {}) /{a } P([,1)} P = {({}, {}, {},{a }) / {a } P[,1)}. Hence the clam. Such study s done n the earler chapters. Thus f Kakutan s theorem s true n the MOD nterval subset matrx specal type of topologcal spaces bult usng [, n) then t s also true n case of some specal type of subspaces of M. a1 a Example.. Let B = { a a a 5 / a P(Z 19 ); 1 5}be the MOD plane subset matrx collecton. Let We wll defne,, n and + operaton on B.

201 MOD Natural Neutrosophc Subset Topologcal x = {(,), (.1,.5),(.,),(1,.1)} {(,.1),(.1,),(.1,)} {(,),(,),(.1,.)} {(,),(1,1),(.,.)} {(1,.1),(.,.)} and y = {(,.1),(.,),(1,5)} {(.1,),(5,.1)} {(,),(,.5)} {(1,.8),(.1,.)} {(1.,),(,.5)} be n B. x y = {(,),(.1,.5),(.,),(1,.1),(1,5)} {(,.1),(.1,),(.1,),(5,.1)} {(,),(,),(.1,.),(,),(,.5)} {(,),(1,1),(.,.),(1,.8),(.1,.)} {(1,.1),(.,.),(1.,),(,.5)} Next we fnd x y; x y = {(.,),(1,.1)} {(.1,)} { } { } { }

202 MOD Subset Topologcal Spaces on the 1 x + y = {(1,.1),(1.1,.),(1.,.1),(,.),(.,),(.,.5)} {(.,),(1.,.1),(1.8),(1.1,5.5),(1.,5),(,5.1)} {(.1,.1),(.,),(.,),(5,.),(5.1,.1),(7.1,.1)} {(7,),(,),(.1,.),(,.5),(,.5),(.1,.5)} {(1,.8),(,1.8),(1.,.8),(.1,.), (1.1,1.),(.,.)} {(.,.1),(1.,.),(,.51),(.,.5)} x n y = {(,.),(.1,.5),(.,),(1,.1), (,),(.,),(.,),(.,), (.15),(.1,.5),(.,),(1.5,)} {(,.),(.1,),(.1,),(,.1), (.5,.),(1.5,)} {(1,),(,),(.,),(8,1.5), (,1),(.,.1)} {(,),(1,.8),(.,.), (.1,.),(.,.1)} {(1.,),(.,),(,.5),(.,.15)} V All the four equatons are dstnct, so all the four semgroups are dstnct and hence all the sx MOD plane subset matrx specal type of topologcal spaces M o = {M,, }, = {M, +, n}, M n M = {M, +, }, M n = {M, +, M n n} and M = {M,, n } are all dstnct. These spaces have subspaces somorphc to S(R n (19)) as well as S([, 19)). However one s not n a poston to say whether Kakutan s theorem s true n case of all these MOD plane subset specal type of topologcal spaces.

203 MOD Natural Neutrosophc Subset Topologcal a1 a Example.7. Let W = { a a / a P(R n (1)); 1 } be the MOD plane subset matrx collecton. (W, +), {W, n }, {W, }, {W, } and {W, } are fve dstnct MOD plane subset matrx semgroups. Let A = {(,.1),(5,.) {(,),(5,), (1,),(1,.1)} (.,)} {(,7),(.1,.) {(,),(,.5), (1,1)} (..,)} B = {(1,.),(,) {(,),(,) (1,.1)} (1,.)} {(,7),(,.), {(,),(8,), (1,),(,5)} (.1,.5)} A + B = {(1,.),(,.5), (,.),(,.), (,.1),(5,.), (1,),(1,.1),(1,.), (,.),(,.1),(,.)} {(,),(.1,7.),(1,8), (,7.),(.1,.),(5,1.), (1,7),(1.1,.),(,),(,), (.1,5.),(1,)} {(,),(5,),(,), (1,.),(.,),(.,) (9,),(1.,.),(,.)} {(,5),(,.5),(.,),(,), (8,.5),(8.,),(.1,.5), (.1,1),(.,.5)}

204 MOD Subset Topologcal Spaces on the A B = {(,.1),(5,.), (1,),(1,.1),(1,.), {(,),(5,),(,), (,),(1,.1)} (.,),(1,.)} {(,7),(.1,.),(1, 1), {(,),(,.5),(.,), (,.),(1,),(,5)} (,),(8,),(.1,.5)} A B = {(1,.1)} {(,)} {(,7)} { } A n B = {(,.),(5,.),(1,), (1,.),(,.),(,.8), {(,),(,8),(1.,) (,),(,.),(,.1), (5,1.),(.,.)} (5,.),(1,.1)} {(,9),(,1.),(,7),(,1.) {(,),(,1),(,), (.,.),(,),(,.), (,),(1.,),(.,1.5), (.1,),(1,),(,5 ),(,1),(,5)} (,.5),(.,)} V Next we fnd A B and B A,

205 MOD Natural Neutrosophc Subset Topologcal {,.1),(5,.),(1,), {(,.1),(5,.),(1,), (1,.1)} {(1,.),(,), (1,.1)} {(,),(,), (1,.1)} {(,),(5,), (1,.)} {(,),(5,), (.,)} {(,7),(,.), (.,)} {(,),(8,), (1,),(,5)} (.1,. 5)} A B = {(,7),(.1,.),(1,1)} {(,7),(.1,.),(1,1)} {(1,.),(,),(1,.1)} {(,),(,),(1,.)} {(,),(,.5),(.,)} {(,),(,.5),(.,)} {(,7),(,.),(1,),(,5)} {(,),(8,),(.1,.5)} = {(,..),(5,.),(1,), (1,.),(,.),(,.8), {(,),(,.),(,.), (,),(,.1),(5,.), (,.),(,.),(5,.), (1,.1) {(,),(,8), (1,),(1,.)} {(,),(,8), (,),(,.8),(1.,.), (1.,),(.5,),(.,)} (5,)} {(,.1),(.1,.),(1,.), {(,),(,),(.,. ), (.8),(,.8),(,),(,.7), (,),(,.1),(.1,.), (.1,.),(1,.1)} {(,1), (1,.)} {(,),(,1), (,.5),(,),(,.), (,),(,),(1.,), (,.1),(.8,),(,), (.,1.5),(,.5), (.,),(,5),(,.5)} (.,)}

206 MOD Subset Topologcal Spaces on the 5 B A = {(,.),(5,.),(1,), (1,.),(,.),(,.8), (,),(,.1),(5,.), (1,.1),(,8.),(5,8.), (1,8),(1,8.),(,8.), (,8.8),(,8),(,8.1), (5,8.),(1,8.1),(,), (,.),(5,.),(1,), (1,.),(,.),(,.8), (,.1),(5,.),(1,.1), (,.8),(5,.8),(1,.8), (1,.8),(,1.),(,1.), (,.8),(,.81),(5,.8), (1,.81),(1.,.), (,.1),(.,.), (.,.),(.,.), (1.,.8),(1., {(,),(,.),(,.), (,.),(,.),(5,.), (1,),(1,.),(,8), (1.,),( 1.), (1.,.),(1.,.1), (.,.),(.,.1), (5,.),(,.),(,), (,.),(5,.),(,.), (5,.8),(5,),(5,.1)}.5,),(.,), (,8.),(,8.),(,8.), (,8.),(5,8.),(1,8), (1,8.),(.5,.),(1.,.), (.5,.),(1.,.), (.5,.),(5.,.), (.5,.),(1.,.), (5.5,.),(.,.), (1.5,),(.,), (1.5,.),(.,.), (1.,),(.,.), (1.,.),(.,.), (.,.),(.,.), (5.,.)}

207 MOD Natural Neutrosophc Subset Topologcal {(,.1),(.1,1.), (1,1.),(,9),(,1.8), (,5),(,1.7),(.1,1.), (1,1.1),(,5.),(.1,.5), (1,.8),(,1.5),(,.), (,7.5),(,.),(.9,.), (.1,.5),(1,.),(1.8,.1), (,.1),(.1,.),(1,.), (,8),(,.8),(,),(,.7), (.1,.),(1,.1),(,.7), (.1,.),(7,.9),(.8.), (,1.),(7,.7),(,.), (,1.),(.1,.),(1.8,.), (,.),(.1,.1),(1,.), (,8.1),(,.9),(,.1), (,.8),(.1,.1),(1,.), (.8,.1),(.9,.),(.8,8), (.8,),(.8,.7),(,5.1), (.1,.),(5,.),(,1),(,.8), (,7),(,.7),(.1,.),(5,.1), (.,.1),(.,.),(1.,.), (.,8),(.,.8),(.,),(.,.7 {(,),(,1),(,), (,), ), (.,.),(1.,.1),(,7.1), (.1,5.),(1,5.),(,),(,5.8), (,9),(,5.7),(.1,5.), (1,5.1),(,.),(.1,.5),(1,.8), (,.5),(,.),(,.5),(,.), (.1,.57),(1,.)} (1.,),(.,1.5), (,.5),(.,), (,),(.,.), (,),(,.1), (.1,.),(1,.), (.,.),(,8), (,8.),(.1,.), (1,.),(,5),(.,1.), (,),(,.1),(.1,1.), (1,1.),(.,),(,), (.,.),(,),(,.1), (.1,.),(.1,.), (1.,.),(,.),(1.,), (.,),(.,),(,.), (5.,),(1.,.1),(1.7,.), (.,.),(.1,.1), (.,5.5),(1,.55), (.8,1.9),(.,.1), (.,.5),(.,.), (.5,1.5),(1.,1.8), (,.5),(.,.5), (,.5),(,.5)}

208 MOD Subset Topologcal Spaces on the 7 Ths s the way operaton s performed on W. Clearly t s left for the reader to verfy A B B A n general. t s easly verfed {W, +}, {W, }, {W, }, {W, n } and {W, } are the fve dstnct MOD plane subset matrx semgroups whch wll yeld W o = {W,, }, W = {W, +, }, W = {W, +, }, W = {W, +, }, W = {W,, }, W n = {W,, }, W = {W, n n, }, W = {W, n n, }, W = {W, n, } and = {W, +, n} the 1 dstnct MOD plane subset W n matrx specal type of topologcal spaces. t s left as open conjecture how many of the MOD plane subset matrx specal type of topologcal spaces satsfy Kakutan s theorem? t has become dffcult to prove Kakutan s theorem n case of these MOD plane subset matrx specal types of topologcal spaces. Next we proceed onto descrbe by examples the MOD plane matrx subset topologcal spaces of specal type. Example.8. Let P(M) = B ={collecton of all subsets from M a1 a = {collecton of all subsets from M = { / a R n () ={(a, b) a a / a, b [,); 1 } be the MOD plane matrx subset collecton. B. We show how operatons +,, and are performed on

209 8 MOD Natural Neutrosophc Subset Topologcal Let (.,1) (5,) A =, (.5,1) (,) (1,1) (,), (,.1) (.5,.) (,) (.1,.1) (.,.7) (.,) and B 1 = (.,1) (5,), (.5,1) (,) (1,1) (.1,.) (,.1) (1,5) be n B. (.,) (,) A + B 1 = {, (1,) (,) (1.,) (1,), (.5,1.1) (.5,.) (.,1) (5.1,.1), (.5,1.7) (.,) (1.,) (5.1,.), (.5,1.1) (1,5) (,) (.1,.), (,.) (1.5,5.) (1,1) (.,.) } (.,.8) (1.,5) (.,1) (5,) A B 1 = { } (.5,1) (,)

210 MOD Subset Topologcal Spaces on the 9 A B 1 = (.,1) (5,), (.5,1) (,) (1,1) (,), (,.1) (.5,.) (,) (.1,.1), (.,.7) (.,) (1,1) (.1,.) (,.1) (1,5) (.9,1) (1,) A n B 1 = { (.5,1) (,), (.,1) (,), (,.1) (,) (,) (.5,), (.1,.7) (,) (.,1) (.5,), (,.1) (,) (1,1) (.,.), (,.1) (.5,.1) (,) (.1,.) } V (,.7) (.,) Clearly,, and V are dstnct. Hence {B, }, {B, }, {B, +} and {B, n } are four dstnct MOD plane matrx subset semgroup. B o = {B,, }, B = {B, n, +}, B = {B,, +}, n B = n {B, B = {B,, n n} and B = {B,, n} are the sx dstnct MOD plane matrx subset specal type of topologcal spaces. Verfyng Kakutan s theorem n case of these MOD plane matrx subset spaces also happens to be a challengng problem for the researchers so left as open conjectures. Example.9. Let P = {P(T)} = {collecton of all subsets from

211 1 MOD Natural Neutrosophc Subset Topologcal a1 a a T = { a a5 a / a R n (7); 1 } be the MOD the MOD plane matrx subset collecton. {P, +},{P, }, {P, } and {P, n be the MOD plane matrx subset semgroups. P o ={P,, }, P = {P, n, +}, P = {P, +, }, n P = {P,, +}, P = {P,, } and P = {P,, } be the MOD plane matrx subset topologcal spaces of specal type. f A = { (,1.1) (1,.5) (.1,.), (,.) (.5,.8) (,.) (,) (1,1) (.1,.7) (.,1) (,.8) (1.,.9), (,) (,5) (.9,1.) (1,) (1.,) (,.) } (,) (,5) (.9,1.) B = { (1,) (1.,) (,.), (1,1) (,) (.1,.8) (.,.) (,) (.9,) } P. (.1,1) (1,.5) (.1,.) We see A B ={ (,.) (.5,.8) (,.), (,) (1,1) (.1,.7) (.,1) (,.8) (1.,.9), (,) (,5) (.9,1.) (1,) (1.,) (,.),

212 MOD Subset Topologcal Spaces on the 11 (1,1) (,) (.1,.8) (.,.) (,) (.9,) } (,) (,5) (.9,1.) A B = { (1,) (1.,) (,.) } Next we proceed onto fnd (,.1) (1,5.5) (.91,1.) A + B = { (5,.) (.1,.8) (,.5), (,) (1,) (1,1.9) (1.,1) (1.,.8) (1.,1.), (,) (,) (1.8,.) (,) (.,) (,.), (1,) (,5) (1,) (1.,.) (1.,) (.9,.), (1,.1) (1,.5) (.11,1) (.,.9) (.5,.8) (.9,.), (1,1) (1,1) (.,1.5) (.8,1.) (,.8) (.1,.9) } We next fnd (,.) (,.5) (.99,.) A n B = { (,) (.8,) (,1.), (,) (,5) (.9,.8) (1.,) (,) (,.7), (,) (,) (.81,1.) (1,) (.5,) (,.9)

213 1 MOD Natural Neutrosophc Subset Topologcal (,1.1) (,) (.1,.1) (.8,.18) (,.8) (.7,), (,) (,) (.1,.5) (.1,.) (,.8) (1.8,) (,) (,) (.9,.9) (.,) (,) (,) } V All the four MOD plane matrx subset semgroups are dstnct as the four equatons,, and V are dstnct. Hence all the sx MOD plane matrx subset specal type of topologcal spaces are dstnct. t s agan left as a open conjecture whether the Kakutan s conjecture s true n case of these MOD plane matrx subset specal type of topologcal spaces. Thus f P(M) = B = {Collecton of all subsets from M = {s t matrces wth entres from R n (m); m <, s, t < }} be the MOD plane matrx subset collecton. Wll the sx MOD plane matrx subset specal type of topologcal spaces B o = {B,, }, = {B, +, n}, B = {B, B n +, }, B = {B, +, }, n B = {B, n n, } and B = {B, n, } satsfy the Kakutan s conjecture or whch of the sx MOD specal type of topologcal spaces satsfy Kakutan s conjecture? Next we proceed onto descrbe by examples MOD plane subset polynomal specal type of topologcal spaces.

214 MOD Subset Topologcal Spaces on the 1 Example.1. Let V = S (P[x]) = {collecton of all subsets from P[x] = { ax / a R n (g)}} be the MOD plane subset polynomal collecton. We can defne four dstnct operatons on V vz. +,, and. Let A = {(,.)x + (1, )x + (.1,.5), (8,.9)x 5 + (., )x + (.1,.1), (1, 1)x + (,.7)x + (.,.9)} and B = {(1,.5)x + (,.5), (, )x + (.1, )x + (,.8)} V. We see A B = {} A B = {(,.)x + (1, )x + (.1,.5), (8,.9)x 5 + (., )x + (.1,.1), (1, 1)x + (,.7)x + (.,.9), (1,.5)x + (,.5), (, )x + (.1, )x + (,.8)} A + B = {(,.)x +(,.5)x + (.1,.1), (8,.9)x 5 + (., ) x + (1,.5)x + (.1,.15), (1, 1)x + (,.7)x + (1,.5)x + (.,.95), (,.)x + (1.1, 5)x + (.1,.85), (8,.9)x 5 + (, )x + (., )x + (.1, )x + (.1,.9), (1, 1)x + (, )x + (,.7)x + (.1, )x + (., 1.7)} A B ={(,.15)x + (1, 1.5)x + (.1,.5)x + (,.15)x + (,.15)x + (,.5), (8,.5)x + (., )x + (.1,.5)x + (,.5)x 5 + (, )x + (,.5), (1,.5)x 5 + (,.5)x + (.,.5)x + (,.5)x + (,.5)x + (,.5), (,.)x + (, )x + (.,.1)x + (,.)x + (.1, )x +(.1,.1)x + (,.)x + (,.)x + (,.), (, 1.8)x 8 + (.8, )x 5 + (.,.)x + (.8, 1.8)x + (., )x + (.1,.)x +(,.7) x 5 + (, )x + (,.8), (, )x 7 + (8, 1.)x 5 + (1., 1.8)x +(.1, )x 5 + (., 1.)x + (., 1.8)x + (,.8)x + (,.5)x + (,.7)} V All the four operatons are dstnct hence the MOD plane polynomal subset semgroups are dstnct. Thus the sx specal

215 1 MOD Natural Neutrosophc Subset Topologcal type of topologcal spaces, V o ={V,, }, V = {V, +, }, V = {V, +, }, {V, +, } = V, V = {V,, } and V = {V,, } are the sx dstnct MOD plane polynomal subset specal type of topologcal spaces of nfnte order. These contan nfnte number of MOD plane subset topologcal zero dvsors and nlpotents however does not contan MOD plane subset topologcal dempotents. Of course V has MOD plane subset strong specal type of topologcal subspaces, however no strong deals. Example.11. Let W = {P(S[x])} = {collecton of all subsets from s[x] = { a x polynomal subset collecton. / a R n ()}} be the MOD plane We can as n the above example defne the four operatons +,, and and {W, +}, {W, }, {W, } and {W, } are four dstnct MOD plane polynomal subset semgroups of nfnte order. Clearly W o = {W,, }, W = {W, +, }, {W, +, } = W, W = {W,, } and W = {W,, } are the MOD plane polynomal subset specal type of topologcal spaces. s the Kakutan theorem true n the case of the sx MOD plane polynomal subset specal type of topologcal spaces? Ths sort s study s very new. n vew of ths we propose the generalzed form of the Kakutan s conjecture n case of MOD plane polynomal subset specal type of topologcal spaces.

216 MOD Subset Topologcal Spaces on the 15 Let S = {P(M[x])} = {collecton of all subsets from M[x] = { ax / a R n (m); m < }} be the MOD plane polynomal subset collecton. {S, +}, {S, }, {S, } and {S, } be the MOD plane polynomal subset semgroups. S o = {S,, }, S = {S, +, }, S = {S, +, }, S = {S, +, }, S = {S,, } and S = {S,, } be the MOD plane polynomal subset specal type of topologcal spaces. n whch of the sx MOD specal type of topologcal polynomal subset spaces the Kakutan s theorem s true. Next we proceed onto descrbe by MOD plane subset coeffcent polynomal specal type of topologcal spaces by examples. Example.1. Let M = { ax / a P(R n (1)} be the MOD plane subset coeffcent polynomal collecton. We can defne +,, and operatons on M. (t s pertnent to keep on record that f p(x) the coeffcent of x term s present and n q(x), the coeffcent x term s not present just say for the sake of argument then n case of unon operaton we just put the {coeffcent as that of x } {} and get the result however n case of operaton we fnd {coeffcent of x } {} =. Ths s the way and ntersecton operatons are performed n M.) We wll llustrate ths stuaton by an example from M. Let p 1 (x) = {(,.1), (, 1), (, 5), (, )}x + {(., ), (, ), (, )}x +{(.1, ), (.11,.), (., ), (1, 1), (.,

217 1 MOD Natural Neutrosophc Subset Topologcal )} and q 1 (x) = {(1, 1), (, ) (,.)}x +{(., ), (, ), (,.)}x +{(, 5), (1, 1), (.1, ), (, ), (, 8)} M. We show how p 1 (x) q 1 (x) and p 1 (x) q 1 (x) are determned. p 1 (x) q 1 (x) = ({(,.1), (, 1), (, 5), (, )} ) x + ({(1, 1), (,), (, )} })x +({(., ), (, ), (,)} {(., ), (, ), (,.)})x + ({(.1, ), (.11,.), (.,), (1, 1), (., )} {(,5), (1,1), (.1, ), (,), (,8)}) = {(,.1), (,1), (,5), (,)}x + {(1,1), (,), (,.)}x +{(., ), (, ), (, ), (,.)}x + {(.1, ), (.11,.), (., ), (1, 1), (., ), (, 5), (, ), (, 8)}. We now fnd p 1 (x) q 1 (x) = ({(,.1), (, 1), (, 5), (, )} )x + (({(1, 1), (, ), (,.)} )x +({(., ), (, ), (, )}) ({(., ), (, ), (,.)})x + ({(.1, ), (.11,.), (., ), (., ), (1, 11)} {(, 5), (1, 1), (.1, ), (, ), (, 8)} = {}x +{}x + {(., ), (, )}x +{(1, 1), (.1, )} = {(., ), (, )}x + {(1, 1), (.1, )}. M). Ths s the way the and operatons are performed on Let p(x) ={(,.), (1, ), (.1,.5)}x +{(1, ), (., )}x +{(, 1), (., 1)} and q(x) ={(.1, ), (.1,.5), (,.)}x +{(.7, ), (1, ), (, 1)}x +{(, 1), (.7,.)} M. p(x) + q(x) ={(.1,.), (1.1, ), (.,.5), (.,.1), (.1,.7), (1.1,.5), (,.), (1,.)}x + {(1.7, 8), (1., ), (,), (1., 8), (1, 7), (., )}x +{(, ), (., ), (.7, 1.), (1, 1.)} p(x) q(x) = {(,.), (1, ), (.1,.5)}x + {(1, ), (., )} x + {(, 1), (., 1)} {(.1, ), (.1,.5), (,.)}x + {(.7, ), (1, ), (, 1)}x +{(, 1), (.7,.)} = ({(,.), (1, ),

218 MOD Subset Topologcal Spaces on the 17 (.1,.5)} {(.1, ), (.1,.5), (,.)})x +({(1, ), (., )} {(.7, ), (1, ), (, 1)})x + ({(, 1), (., 1)} {(, 1), (.7,.)}) = ({(,.) (1, ), (.1,.5), (.1, ))x +{(1, ), (., ), (.7, )}x + {(, 1), (., 1), (.7,.)} p(x) q(x) = ({(,.), (1, ), (.1,.5)} {(.1, ), (.1,.5), (,.)})x + ({(1, ), (., )} {(.7, ), (1, )}) x + ({(, 1), (., 1)} {(, 1), (.7,.)}) = {(,.), (.1,.5)}x +{(1, )}x + {(, 1)} We now fnd p(x) q(x) = ({(,.), (1, ), (.1,.5)} {(.1, ), (.1,.5), (,.)})x + ({(,.), (1, ), (.1,.5)} {(.7, ), (1, ), (, 1)})x + ({(,.), (1,), (.1,.5)} {(, 1), (.7,.)})x + ({(1, ), (., )} {(.1, ), (.1,.5), (,.)})x + ({(1, ), (., )} {(.7, ), (1, ), (.1)})x + ({(1, ), (., )} {(, 1), (.7,.)})x + ({((, 1), (., 1)} {(.1, ), (.1,.5), (,.)})x +({(, 1), (., 1)} {(.7, ), (1, ), (, 1)})x + {(, 1), (., 1)} {(, 1), (.7,.)} = {(, ), (.1, ), (.1, ), (,.1), (.1,.5), (,.)}x + {(,.), (.7, ), (.7,.1), (, 1.), (1, ), (.1, ), (,.), (, ), (,.5)}x + {(,.), (, ), (,.5), (,.), (.7, ), (.7,.1)}x + {(.1,), (.1, ), (, 1.), (., ), (., 1), (,.)}x +{(.7, ), (1, ), (, ), (., ), (., ), (, )}x + {(, ), (.7, 1), (, ), (.,.)}x + {(, ), (,.5), (,.), (., ), (.,.5), (,.)}x + {(, ), (, ), (, 1), (.1, ), (., )}x + {(, 1), (,.), (.1,.)} V All the four equatons,, and V are dstnct. All the four values are dstnct so {M, }, {M, }, {M, } and {M, } are the MOD plane subset coeffcent polynomal semgroups. Hence the sx MOD plane subset coeffcent polynomal specal type of topologcal spaces M o = {M,, }, M = {M,

219 18 MOD Natural Neutrosophc Subset Topologcal +, }, M = {M, +, }, M = {M,, }, M = {M,, } are dstnct. M = {M, +, } and t s a open conjecture to verfy the valdty of the Kakutan s theorem n case of ths M. Example.1. Let S = { a x / a P(R n (11))} be the MOD plane subset coeffcent polynomal collecton. {S, +}, {S, }, {S, } and {S, } be the MOD plane subset coeffcent semgroup. S o = {S,, }, S = {S, +, }, S = {S, +, }, S = {S, +, }, S = {S,, } and S = {S,, } are the MOD plane subset coeffcent polynomal specal type of topologcal spaces all of whch are dstnct. t s a challengng problem to study whether the Kakutan s theorem s true n case of all these sx spaces or only some of them. We propose the followng conjecture. Conjecture.7. Let S = { a x / a P(R n (m)), m < } be the MOD plane subset coeffcent polynomal collecton. Whch of the sx specal type of topologcal spaces usng MOD plane subset coeffcent polynomals, S o = {S,, }, S = {S, +, }, S = {S, +, }, S = {S, +, }, S = {S,, } and S = {S,, } satsfy Kakutan s theorem? Ths s not only a challengng and dffcult one. Thus t s not easy to fnd a soluton however can atleast try to prove Kakutan s theorem s not true n these cases.

220 MOD Subset Topologcal Spaces on the 19 Next we provde examples of MOD plane polynomal subsets of fnte degree. Example.1. Let S = P(M[x] 1 ) = {collecton of all subsets 1 from M[x] 1 = { ax / a R n (); x 1 = 1}} be the MOD plane fnte degree polynomal subset collecton. We can have four operaton +,, and on S and are dstnct. We just ndcate the stuaton. Let A ={(.7, )x + (,.1)x + (, ), (1,.)x + (1,.1), (1,.1)x + (., )} and B = {(1,.1)x + (., ), (.7, )x + (., 1)x + (.1,.)} S. A B = {(.7, )x + (,.1)x + (, ), (1,.)x + (1,.1), (1,.1)x + (., ), (.7, )x + (., 1)x + (.1,.)} A B = {(1,.1)x + (., )} A + B = {(.7, )x +(,.11)x + (., ), (1,.)x + (1,.1)x + (1.,.1), (,.)x + (., ), (.7, )x + (.7, )x + (., 1)x + (,.1)x + (.1,.), (.7, )x +(1., 1.)x + (1.1,.1), (.7,)x + (., 1)x + (1,.1)x +(.,.)} A B ={(.7, )x + (,.1)x + (,.)x + (.1, )x + (., )x, (1,.)x + (1,.1)x + (., )x + (.,.1), (1,.1)x + (., )x + (., ), (.9,)x 7 + (1.,.)x 5 + (, )x + (.1, )x 5 + (.,.1)x + (, )x + (.7, )x + (.,.)x + (,.), (.7,.)x + (.7,.)x + (.,.)x + (.,.1)x + (.1,.)x + (.1,.), (.7,.)x 5 + (.1, )x + (.,.1)x + (., )x + (.1,.)x + (., )} V All the four operatons are dstnct hence the MOD plane fnte degree polynomal subset semgroups {S, +}, {S, }, {S,

221 MOD Natural Neutrosophc Subset Topologcal } and {S, } are dstnct. Hence the MOD plane specal type of topologcal spaces S o = {S,, }, S = {S,, +}, S = {S, +, }, S = {S,, +}, S = {S,, } and S = {S,, } are all dstnct. However one s not n a poston to predct whether the Kakutan s theorem s true or not n these cases. Example.15. Let M = {P(S[M] 1 )} = {collecton of all subsets from S[M] 1 = { 1 a x / a R n (7); x 11 = 1}} be the MOD plane fnte degree polynomal subset collecton {M, +}, (M, }, {M, } and {M, } are the MOD plane fnte degree polynomal subset semgroups. Let M o = {M,, }, {M, +, } = M, M = {M, +, }, M = {M, +, }, M = {M,, } and M = {M,, } be the MOD plane fnte degree polynomal subset specal type of topologcal spaces. Verfy Kakutan s theorem for these spaces. n vew of all these we suggest the followng conjecture. Conjecture.8. Let W = {P(S[x] t )} = {collecton of all subsets t from S[x] t = { ax / a R n (m); x t+1 = 1; t <, m < } be the MOD plane fnte degree polynomal subset collecton. {W, }, {W, }, {W, +} and {W, } be the MOD plane fnte degree polynomal subset semgroup. W o = {W,, }, W = {W,, +}, W = {W, +, }, W = {W, +, }, W = {W,, } and W ={W,, } be the

222 MOD Subset Topologcal Spaces on the 1 MOD plane fnte degree polynomal subset specal type of topologcal spaces. Verfy the valdty of Kakutan s theorem n case of these sx spaces. Next we proceed onto descrbe by examples the MOD plane subset coeffcent fnte degree polynomal specal type of topologcal spaces. Example.1. Let V = S[x] 19 = { ax /a P(R n (8)); x = 1} be the collecton of all MOD plane subset coeffcent fnte degree polynomal collecton. On V we can defne the four operatons,, + and and all the four operatons are dstnct. Let p(x) = {(,.), (,.), (7, 1), (1, 1)}x +{(,.1), (, ), (., 1)}x + {(.7, ), (1, 1), (,.)}x + {(, ), (7,.5), (1,.1)} and q(x) ={(,.1), (1, 1), (., 1), (1, )}x + {(.7, ), (1, 1), (,.), (1, ), (,.)}x + {, ), (7,.5), (1,.1), (, ), (.1,.)} V. 19 q(x). We fne p(x) + q(x), p(x) q(x), p(x) q(x) and p(x) p(x) + q(x) = {(,.), (,.7), (1, 1.1), (, 1.1), (1, 1.), (5, 1.), (, ), (, ), (., 1.), (., 1.), (7., ), (1., ), (1,.), (5,.), (, 1), (, 1)}x + {(,.1), (, ), (., 1)}x + {(1.,), (1.7,1), (.7,.), (, ), (7, 1.), (.7,.), (, 1.), (1,.9), (1.7, ), (, ), (7,.), (.7,.), (7, 1.), (, 1)}x + {(, ), (7,.5), (1,.1), (7,.5), (, 1), (,.), (1,.1), (,.), (,.), (, ), (1,.5), (,.1), (7.1,.7), (1.1,.), (.1,.)} p(x) q(x) = {(,.), (,.), (7, 1), (1.1), (.1), (., 1), (1, )}x + {(,.1), (, ), (., 1)}x + {(.7, ), (1, 1), (,

223 MOD Natural Neutrosophc Subset Topologcal.), (,.), (1, ), (,.)}x + {, ), (7,.5), (1,.1), (, ), (.1, )} p(x) q(x) = {(1,1)}x + {}x + {(.7, ), (1, 1)}x + {(, ), (7,.5), (1,.1)} p(x) q(x) = {(,.), (1,.), (,.1), (,.1), (,.), (,.), (7, 1), (1, 1), (,.), (1.,.), (.8, 1), (., 1), (, ), (, ), (7, ), (1, )}x + {(,.1), (,), (.,.1), (,.1), (., 1), (.,.1), (.1, 1), (, ), (., )}x 5 +{(1., ), (,.1), (,.), (.7, ), (1, 1), (,.), (.8, ), (., 1), (.,.), (.7, ), (1, ), (, )}x + {(,.), (,.5), (,.1), (, ), (7,.5), (1,.1), (.8,.5), (.,.1), (, ), (7,.5), (1, )}x + {(, ), (.8, ), (.9, ), (.7, ), (,.), (,.), (7, 1), (1 1), (,.), (,.18), (5,.), (,.), (,.), (, 1.), (7, ), (1, ), (,.8), (,.), (,.), (,.)}x + {(., ), (, ), (.1, ), (,.1), (., 1), (,.), (.9,.)}, (,.), (., ), (,.), (1.8,.)}x + {(.9, ), (.7, ), (., ), (.7, ), (1, 1), (,.), (.1, ), (,.), (,.18), (1, ), (, 1.), (,.), (., ), (,.)}x + {(, ), (.9, ), (.7, ), (, ), (7,.5), (1,.1), (.1, ), (,.9), (5,.15), (, ), (7, 1), (1,.), (,.1), (,.), (,.)}x + {(,.), (, 1.8), (, ), (,.1), (,.), (, 1.8), (1,.5), (7,.5), (,.), (,.), (, ), (7,.1), (1,.1), (,.), (, ), (,.), (.,.1), (.7,.), (.1,.)}x +{(,.), (, ), (, ), (,.5), (.1,.5), (,.1), (.,.1), (,.), (., ), (.,.), (.,.1)}x + {(, ), (, ), (,.18), (.9, ), (7,.5), (,.), (.7, ), (1,.1), (,.), (, ), (, ), (, 1.8), (.7, ), (.1,.), (.,.1)}x +{(, 9), (,1.5), (,.), (1,.5), (7,.5), (,.), (7,.5), (1,.1), (, 1), (, 1.5), (,.)} V We see all the four equatons are dstnct so the four MOD plane subset coeffcent polynomal semgroups, are {V, +}, {V, }, {V, } and {V, } are dstnct. Thus the MOD plane subset coeffcent polynomal specal type of topologcal spaces;

224 MOD Subset Topologcal Spaces on the V o = {V,, }, V = {V, +, }, V = {V, +, }, V = {V, +, }, V = {V,, } and V = {V,, } are all dstnct. Now the valdty whether these sx spaces satsfy Kakutan s theorem s challengng work. Ths s left as a open conjecture for the reader. Example.17. Let M = { 9 a x / a (R n (1)), x 1 = 1} be the MOD plane fnte degree polynomal subset coeffcent collecton {M, }, {M, }, {M, +} and {M, } be the MOD plane fnte degree polynomal subset coeffcent semgroup. M o = {M,, }, M = {M, +, }, M = {M,, }, M = {M,, }, M ={M, +, } and M = {M, +, } be the MOD plane fnte degree polynomal wth subset coeffcent specal type of topologcal spaces. Study the valdty of Kakutan s theorem n case of these sx specal types of topologcal spaces. n vew of ths we propose the followng conjecture. Conjecture.9. Let t S = { ax / a P(R n (m)), t, m <, x t+1 = 1} be the MOD plane fnte degree polynomal wth subset coeffcents from R n (m); S o = {S,, }, S ={S, +, }, S = {S, +, },S = {S, +, }, S = {S,, } and S = {S,, } be the MOD plane fnte degree polynomal subset coeffcent specal type of topologcal spaces.

225 MOD Natural Neutrosophc Subset Topologcal Prove or dsprove the valdty of Kakutan s theorem n the case of the above specal type of topologcal spaces. However n ths chapter we proceed onto dscuss how far Kakutan s theorem can be studed n case of real neutrosophc space R = {a + b / a, b R, = }, the real dual number plane R g = {a + bg / a, b R; g = }, the real specal dual lke number space R h = {a + bh / a, b R, h = h} and the real specal quas dual number space R k = {a + bk / a, b R, k = k}. All these four spaces are not lke the real planes. These are planes whch enjoy dfferent knd of propertes. However n all these planes f some one wshes to defne a meanngful functon or map f they must undoubtedly have at least a fxed pont gven by f() = n case of real neutrosophc plane, f(g) = g s case of real dual number plane, f(h) = h n case of specal dual lke number plane and f(k) = k s case of real specal quas dual number plane. Thus all these are partally real planes or to be more techncal semreal planes. Subset specal type of topologcal spaces have been defned usng them [ ] now our contenton s can we have the modfed form of Kakutan s theorem or to be more specfc wll Kakutan s theorem be true on these spaces uncondtonally. Study n ths drecton s new for all these spaces demand a fxed pont theorem for any nonempty subset S of R R n tmes any set valued functon : S S then wll S be a fxed pont of? (for any arbtrary subset S of (R) n.

226 MOD Subset Topologcal Spaces on the 5 Smlar conjecture n case of R g, R h and R k. However these are also planes how best s the Kakutan s fxed pont theorem true n case of specal type of topologcal spaces bult usng these neutrosophc plane R, the dual number plane R g, the specal dual lke number plane R h and the specal quas dual number plane R k. Conjecture.1. Let R the reals n Kakutan s fxed pont theorem be replaced by R or R g or R h or R k. Can Kakutan fxed pont theorem hold n these four cases? Conjecture.11. Let specal subset topologcal spaces be constructed usng R or R k or R h or R g; wll Kakutan s theorem be true on these topologcal spaces? n authors vew the proof of the Kakutan s theorem n case of R g, R, R k and R h happen to be a routne task wth approprate modfcatons by usng only classcal or usual topology on R g, R, R k and R h. However n case of specal type of subset topologcal space the theorem happens to be dffcult. Next we proceed onto study the valdty of Kakutan s fxed pont theorem n case of the secton of real plane R n (m); m < where R n (m) = {(a, b) / b, a [, m)}. Ths s dscussed earler. Now the noton of MOD plane topologcal spaces of specal type s dscussed for the valdty of Kakutan s theorem. Now n earler chapter we have ntroduced MOD nterval neutrosophc specal type of topologcal spaces usng [, n) = {a + b / a, b [, n)}, n <, MOD nterval; dual number specal type of topologcal spaces usng [, n) g = {a + bg / a, b [, n), g = }; n < and MOD nterval specal dual lke number specal type of topologcal spaces usng [, n) h = {a + bh / a, b [, n); h = h}; n <,

227 MOD Natural Neutrosophc Subset Topologcal MOD nterval quas dual number specal type of topologcal spaces usng [, n) k = {a + bk / a, b [, n), k = (n 1) k}. Ths study s nterestng and nnovatve for the MOD nterval s a segment of the specfc planes cut from the frst quadrant of the respectve planes. nfact t s a msnomer we call t as MOD nterval for the planes are sketched below for the reader to make a comparatve study. m R [,m) 1 1 m x Fgure. Shaded regon s the MOD neutrosophc nterval or the MOD neutrosophc plane. R = {a + b / a, b R, = } [, m) = {a + b / a, b [, m); m <, = } Thus [, m) R and t exactly occupes part of the frst quadrant of the neutrosophc plane R.

228 MOD Subset Topologcal Spaces on the 7 Smlarly mg Rg g [,m)g g 1 m Fgure. T he dual number plane s Rg ={a + bg / a, b R, g = } and the MOD dual number nterval s [, m) g = {a + bg / a, b [, m), m <, g = } s the MOD dual number plane shown by the shaded regon.clearly [, m) g R g, thus the MOD dual number plane s a subset (subplane) of R g occupyng some secton of the frst quadrant of R g. Now consder the real specal dual lke number plane gven by the followng R h = {a + bh / a, b R, h = h} mh Rh h [,m)h h h 1 m Fgure.

229 8 MOD Natural Neutrosophc Subset Topologcal Now the MOD specal dual lke number nterval / plane s [, m) h = {a + bh / a, b [, m), h = h} s shown by the shaded regon. nfact [, m) h R h thus [, m) h s a subplane of R h. Smlarly R k = {a + bk /a, b R; k = k} be the real specal quas dual plane. [, m) k ={a + bk / a, b [, m), k = (m 1)k} be the MOD specal quas dual number plane. The representaton of them s as follows. mk Rk k [,m)k k 1 k 1 m k Fgure.5 Here also t s seen the plane R k contans the MOD specal quas dual number plane [, m) k; that s shown by the shaded area n the fgure. Thus [, m) k R k. Fnally we see C([, m)) = C(m) = {a + b F / a,b [, m), F = (m 1)} s the MOD nterval fnte complex numbers. Clearly ths also s a substructure but t s not extendable as n case of = 1 and F = (m 1) wth lmtatons nfact all contanment has lmtatons for (m 1) = 1 (mod m) n all cases whch s untrue n R.

230 MOD Subset Topologcal Spaces on the 9 Thus t s only as subset for none of the algebrac propertes or analytc propertes are true n ths case. Yet wth these lmtatons we have the followng contanment relaton C([,n)) = C(n) C = {a + b / = 1 a, b R} C m F F C(n) = C([, n)) F 1 m Fgure. Thus all the four planes are only contaned as respectve subset. n case of complex plane and MOD fnte complex number plane even ths sort of contanment s not true for we have to dentfy wth F that s F then only such result s true a lttle devant from other three planes for we see n case of real specal quas dual number plane, k = k and n MOD specal quas dual number plane [, m) k, k = (m 1)k. Now havng seen the relatve structure t s pertnent to keep on record there s lots of dfference n ther algebrac structures for the product on them can gve zero dvsors.

231 MOD Natural Neutrosophc Subset Topologcal Thus wth these devatons n mnd only we buld the topologcal structures on them. We see the MOD nterval topologcal spaces bult usng S = {[, n) g}, H = {[, n) h}, J ={[, n) k}, L = {C([,n))} and M = {[, n) } be the fve MOD nterval / planes and let A = P(S), B = P(H), C = P(J), D = P(L) and E = P(M) be the collecton of all subsets of the fve MOD nterval planes respectvely. We can have the sx types of MOD plane subset topologcal spaces bult on each of them. t s conjectured whch of these MOD plane specal type of topologcal spaces satsfy the Kakutan fxed pont theorem? So study n ths drecton s suggested as open conjectures as one s not n a poston to know the propertes of these MOD plane / nterval spaces as well as planes bult on R, R g, R h and R k. Smlarly study or adaptaton of Kakutan s fxed pont theorem n case of MOD fnte complex number nterval / planes usng P(C[, n)) happens to be a open conjecture. Thus all the spaces whether n the frst place satsfy the norms of a Kakutan s theorem s to be analysed. Further a functon yeldng convex graph s not feasble n these cases for even functons yeldng smple smooth curves happen to become dscontnuous n MOD planes, so study n ths drecton s dffcult. Next we study the propertes of fnte topologcal spaces bult usng MOD natural neutrosophc sets vz; Z n, C (Z n ), Z n, Z n g, Z n h and Z n k.

232 MOD Subset Topologcal Spaces on the 1 We can fnd the MOD natural neutrosophc subset specal type of topologcal spaces got from P( Z ), P(C (Z n )), P(Z n ), P(Z n g, P(Z n h ) and P(Z n k ). Such spaces have been ntroduced earler. The followng s conjecture s ntroduced. Conjecture.1. Let A = P( Z ), B = P(C (Z n )), C = P((Z n n )) D = P(Z n g ), E = P(Z n h ) and F = P(Z n k ) be the MOD natural neutrosophc collecton of subsets. Let the MOD natural neutrosophc subset specal type of topologcal spaces be bult on A, B, C, D, E and F. s Kakutan fxed pont theorem true n these cases? Next we see one example to llustrate how n these spaces one should always have certan elements to be fxed. Example.18. Let V = P ( Z ) = {P({, 1,,,,, +, + 1, +, +, + collecton. lkewse n +, +1, +, +, + + 1, + + })} be the MOD natural neutrosophc subset Here we see cannot be mapped to any element but should be mapped only onto. Now f V o = {V,, } s the MOD natural neutrosophc specal type of subset topologcal space., 1. s the Kakutan s fxed pont theorem true n case of V o?. s the condtons for Kakutan s theorem true n case of V o?

233 MOD Natural Neutrosophc Subset Topologcal Study questons (1) and () n case of V = {V, +, }, V = {V, +, }, V = {V, +, }, V = {V,, } and V = {V,, }. Now we can defne on V max and mn operaton by takng face values. max {, } =, max {, a} = a f we choose to keep n our study real value to be domnatng so mn {, a} =. n a smlar fashon f we want the MOD natural neutrosophc value domnatng we put max {, a} = and mn {, a} = a. t s upto the reader to fx mn {, } = and max{, } = usng the face value orderng. So under these crcumstances {V, mn, max} by face value orderng s a MOD natural neutrosophc subset specal type of topologcal spaces. Check the valdty of the Kakutan s fxed pont theorem for ths V. Can we use ths to model Nash equlbrum n game theory where one can also say the predctons are not determnable that s an ndetermnate or natural neutrosophc element n the predcton of the wnner. For n all cases of MOD natural neutrosophc elements we propose the followng problem. Can the fxed pont exst even f other condtons of the Kakutan s theorem s not satsfed by the MOD topologcal spaces? Example.19. Let V ={P(Z 9 g ) = {collecton of all subsets from Z 9 g = {a + bg + / a, b Z 9, g =, t Z 9 s such g t

234 MOD Subset Topologcal Spaces on the that t s a nlpotent or a zero dvsor or an dempotent n Z 9 }} be the MOD natural neutrosophc dual number subsets collecton. V o, V, V, V, V and V be the MOD natural neutrosophc dual number subsets specal type of topologcal space. Prove or dsprove there s a fxed pont for these spaces mmateral of the Kakutan other propertes beng true. Example.. Let T = {P(Z 8 h )} = {collecton of all h subsets from Z 8 h = {a + bh + t / a, b Z 8, h = h, t Z 8 and t s nlpotent or zero-dvsor or dempotent of Z 8 }} be the MOD natural neutrosophc specal dual lke number subset collecton. T o, T, T, T, T and T be the MOD natural neutrosophc specal dual lke number subset specal type of topologcal spaces. The reader s left wth the task to fnd fxed ponts and prove or dsprove the space s connected or compact. Example.1. Let F = {P(Z 1 k )} ={collecton of all k subsets from Z 5 k = {a + bk + t / a, b Z 1, k = k, t s an dempotent or nlpotent or a zero dvsor n Z 1 } be the MOD natural neutrosophc specal quas dual number subset collecton. F o, F, F, F, F and F be the MOD natural neutrosophc specal quas dual number subset specal type of topologcal spaces. Can ths have Kakutan s fxed pont? Prove or dsprove these topologcal spaces are compact / connected. Example.. Let G = {P(C (Z )} ={collecton of all subsets from C c (Z ) ={a + b F + t / a, b Z, F =, t Z s an dempotent or a zero dvsor or a nlpotent element n Z }} be the MOD natural neutrosophc fnte complex number subset collecton.

235 MOD Natural Neutrosophc Subset Topologcal Let G o, G, G, G, G and G be MOD natural neutrosophc fnte complex number subset specal type of topologcal spaces. Study the propertes of Kakutan s theorem for these topologcal spaces bult usng G. Example.. Let H ={P(Z g )} = {collecton of all g subsets from Z g = {a + bg + t / a, b Z, g =, t Z s such that t s an dempotent or a zero dvsor or a nlpotent}} be the MOD natural neutrosophc dual number subset collecton. H o, H, H, H, H and H be the MOD natural neutrosophc dual number specal type of topologcal spaces. Verfy Kakutan s theorem for these spaces. Example.. Let ={P(Z 1 h)} = {collecton of all subsets h from Z 1 h ={a + bh + t / a, b Z 1, h = h, t Z 1 s such that t s a nlpotent or an dempotent or a zero dvsor}} be the MOD natural neutrosophc specal dual lke number subset collecton. Study whether fxed pont can be acheved for the spaces o,,,, and. Lkewse we can study usng MOD natural neutrosophc specal quas dual number subsets collecton from P(Z n k ) ={collecton of all subsets from Z n k = {a + bk + / a, b Z n, k = (n 1)k, t Z n s such that t s a nlpotent or an dempotent or a zero dvsor n Z n. Study the topologcal spaces bult usng D for Kakutan s theorem. Lkewse study for E = P(C (Z n )) ={collecton of subsets from C c (Z n ) = {a + b F + t / a, b Z n, F = (n 1); t n Z n s such that t s a nlpotent or a zero dvsor or an dempotent. k t

236 MOD Subset Topologcal Spaces on the 5 All these study s a matter of routne. Further buldng matrces wth subsets from P( Z ), P(Z n g ), P(Z n h ), P(Z n k, P(Z n and P(C (Z n )) can be carred out and one can seek for the fxed pont by modfyng the Kakutan s theorem. Fnally one can also take collecton of matrx subsets wth entres from Z n, C (Z n ), Z n g, Z n h, Z n k and Z n and obtan fxed ponts whether other condtons of Kakutan s theorem s true or not. n Lkewse one can get MOD natural neutrosophc number specal type of topologcal spaces usng subsets of { P( Z n )} or { ax / a P(C (z n ))} or { ax /a ([, n) g )} or { ax / a P([, n) h )} or { n) k }. a x a x / a / a P([, Can we have fxed pont rulng out the other propertes of Kakutan s theorem? Next we can study the exstence of fxed pont wth or wthout the other propertes of Kakutan s theorem for the followng MOD natural neutrosophc fnte degree polynomals wth subset quotent specal type of topologcal spaces bult m usng S = { ax / a P( Z n ), x m+1 = 1} or P( Z n ) replaced by P(Z n ) or P(C (Z n )) or P(Z n g ) or P(Z n h ) or P(Z n k ). We can study smlarly for the exstence of fxed ponts or modfed form of Kakutan s theorem n case of (S[x] m ) =

237 MOD Natural Neutrosophc Subset Topologcal {collecton of all subsets from S[x] m = { ax / x m+1 = 1, a P( Z n )} = P the MOD natural neutrosophc specal type of fnte degree polynomal subsets topologcal spaces. We also suggest the study of Z n replaced by C (Z n ) or Z n k or Z n h or Z n g or Z n. For n all these spaces one can easly prove the exstence of fxed ponts whether the other condtons of Kakutan s theorem s true or not. Next we carry out the study on MOD nterval natural neutrosophc number subsets whch wll be llustrated by some examples. Example.5. Let S ={P( [, 8))} = {collecton of all subsets from 8 [, 8) = {a + t / a [, 8) and t s an dempotent or nlpotent or a zero dvsor n Z 8 } = collecton of all MOD nterval natural neutrosophc subsets. S o, S, S, S, S and S be the MOD nterval natural neutrosophc subset specal type of topologcal spaces. Prove these have MOD natural neutrosophc fxed ponts even f the other propertes of Kakutan s theorem s not true. Example.. Let R = {P(C ([, ))} = {collecton of all subsets from C c ([, )) ={a + b F + t / a, b [, ), =, F t Z s an dempotent or a nlpotent or an dempotent n Z } be the MOD nterval natural neutrosophc fnte complex number subsets collecton. R o, R, R, R, R and R be the MOD nterval natural neutrosophc fnte complex subsets specal type of topologcal spaces. m

238 MOD Subset Topologcal Spaces on the 7 Prove these topologcal spaces have fxed pont. Further prove or dsprove these spaces satsfy the other propertes of Kakutan s theorem. Smlar study can be done usng P([, n) g ) or P([, n) h ) or P([, n) k ) or P([, n) ) as MOD nterval neutrosophc subset collectons ther respectve topologcal spaces can be analysed for fxed pont as well as about the valdty of Kakutan s theorem. a1 a Example.7. Let B = { a / a P(C ([, ))) = {collecton a a 5 of all subsets from C c ([, )) = {a + b F + t / a, b [, ), F =19, t Z s an dempotent or a nlpotent or a zero dvsor? MOD nterval natural neutrosophc fnte complex number matrx wth subset entres collecton. B o, B, B, n B, n B and n B be the MOD nterval fnte complex number natural neutrosophc matrx wth subset entres specal type of topologcal spaces. Prove these spaces have fxed pont and further verfy whether all other propertes of Kakutan s theorem are true. Study or prove the exstence of fxed pont when MOD nterval natural neutrosophc spaces are constructed usng subsets from P([, n) g ) or P(([, n) h ) or P( [, n)) or P([, n) k ) or P([, n) ). Further can for these spaces, all the propertes of Kakutan s theorem hold good? Example.8. Let A = P(M) = {collecton of all subsets from

239 8 MOD Natural Neutrosophc Subset Topologcal a a a M = { a a a / a [, 9) h h ={a + bh + t /a, b [, 9), h = h, t Z 9 s a nlpotent or an dempotent or a zero dvsor n Z 9, 1 1}} be the MOD nterval natural neutrosophc specal dual lke number matrx subset collecton. A o, A, A, n A, n A and n A be the MOD nterval natural neutrosophc specal dual lke number matrx subset specal type of topologcal spaces. Prove these spaces have fxed pont but however test for the other propertes of Kakutan s theorem. Study the above example by replacng [, 9) by [, 9) or [, g or [, 15) k or C ([, )) for the fxed pont of the respectve MOD topologcal spaces and the valdty of the Kakutan s theorem. Example.9. Let V = { m a x / a P([, 8) g )} be the MOD nterval natural neutrosophc dual number subset coeffcent polynomal collecton. V o, V, V, V, V and V be MOD nterval natural neutrosophc dual number subset coeffcent polynomal specal type of topologcal spaces. Prove these spaces have fxed pont. Test the valdty of other propertes of Kakutan s theorem satsfed by these spaces. Study the above example when P([, 8) g ) s replaced by P( [, )) or P([, 1) h ) or P(C ([, 8)) or P([, 5) k ) orp([, 1) ) and the related MOD nterval natural neutrosophc subset coeffcent polynomal topologcal spaces for fxed pont and other propertes of Kakutan s theorem.

240 MOD Subset Topologcal Spaces on the 9 Example.. Let M = P(S[x]) = {collecton of all subsets from S[x] = { a x / a [, 11) } be the MOD nterval natural neutrosophc-neutrosophc polynomal subset collecton. M o, M, M, M, M and M be the MOD nterval natural neutrosophc-neutrosophc polynomal subset specal type of topologcal spaces. Prove all these spaces have fxed pont and verfy other propertes of Kakutan s theorem for these spaces. n example. replace [, 11) by [, 8), [, 18) g, [, 5) h, [,1) k and C ([, 7)) and study the respectve MOD nterval natural neutrosophc polynomal subsets specal type of topologcal spaces for fxed pont and other propertes of Kakutan s theorem. Example.1. Let W = {(S[x] 1 ) = {collecton of all subsets from S[x] 1 = { 1 a x / x 11 =1, a [,1) k }} be the MOD nterval natural neutrosophc specal quas dual number fnte degree polynomal subset collecton. W o, W, W, W, W and W be the MOD nterval natural neutrosophc specal quas dual number fnte degree polynomal subset specal type of topologcal spaces. Study Kakutan s fxed pont theorem for these spaces. n W of example.1, replace [, 1) k by C ([, )), [, 15) g, [, 8) h, [, 18) and study ther respectve MOD nterval natural neutrosophc fnte degree polynomal subset specal type of topologcal spaces for Kakutan s theorem.

241 MOD Natural Neutrosophc Subset Topologcal 9 Example.. Let N = { ax / a P([, 7) h ); x 1 = 1} be the MOD neutrosophc specal dual lke number subset coeffcent of fnte degree polynomal subset collecton. N o, N, N, N, N and N be the MOD nterval natural neutrosophc specal dual lke number specal type of topologcal spaces. Study these spaces for the valdty of the Kakutan s theorem. n the above example replace P([, 7) h by P( [, 8)), P([, 8) k ), P([,7) g ), P([, 1) ) and P(C ([,)) and study ther respectve MOD nterval natural neutrosophc fnte degree polynomals wth subset coeffcents for the valdty of the Kakutan s theorem. We have suggested several problems for ths chapter some of whch are at research level. Further the study of valdty of Kakutan s theorem happens to be a dffcult research for there are fxed ponts n MOD topologcal spaces wth or wthout satsfyng the other propertes of Kakutan s theorem. PROBLEMS 1. Obtan all the specal propertes enjoyed by MOD real planes R n (m) = {[, m) [, m) = {(a, b) / a, b [, m), m < }}.. Prove {R n (m), +} s an abelan group of nfnte order.

242 MOD Subset Topologcal Spaces on the 1. Prove {R n (m), +} can have both fnte and nfnte order subgroups.. Prove {R n (m), } s a MOD plane semgroup under the operaton. 5. Prove {R n (m), } has zero dvsors whch are nfnte n number.. Prove {R n (m), } can have dempotents f and only f Z m has dempotents. 7. Can {R n (m), } have nlpotents? Justfy your clam. 8. Prove {R n (m), +, } s a MOD plane rng. Study specal features assocated wth t. 9. Let S ={R n (), +, } be the MOD plane rng. ) Does S n general satsfy the dstrbutve law? ) Prove o(s) =. ) Fnd all MOD plane zero dvsors of S. v) Prove S has MOD plane dempotents. v) Prove S has MOD nlpotents. v) Fnd all fnte order subrngs of S. v) v) Prove all deals of S are of nfnte order. Obtan any other specal feature assocated wth S.

243 MOD Natural Neutrosophc Subset Topologcal 1. Let M = {R n (19); +, } be the MOD plane rng. ) Study questons () to (v) of problem (9). ) Dstngush ths M from S of problem (9). 11. Let T = P(R n (1)) = {collecton of all subsets from R n (1)} be the MOD plane subsets collecton. ) Prove {T, +}, {T, }, {T, } and {T, } are MOD plane semgroups whch are dstnct. ) ) v) Prove {T, } s a semgroup of nfnte order wth nfnte number of zero dvsors. Prove T o = {T,, }, T = {T, +, }, T = {T, +, }, T n = {T, +, }, T = {T, n, } n and T ={T, n, } are sx dstnct MOD plane subset specal type of topologcal spaces. Whch of these spaces are compact? v) Are these spaces connected? v) v) v) x) Can these spaces satsfy the Kakutan s fxed pont theorem? Prove these MOD plane topologcal spaces can have subspaces of fnte order. Prove all MOD plane subset topologcal deals are of nfnte order. Can T have MOD plane strong topologcal subspaces?

244 MOD Subset Topologcal Spaces on the x) Prove T can never have MOD plane strong topologcal deals? x) x) x) xv) Prove T can never have MOD plane topologcal deals? Can T have MOD plane topologcal ¾ th strong deals? Can T have MOD plane topologcal ½ strong deals? Obtan any other specal feature assocated wth T. 1. Let B = {P(R n (18)} = {collecton of all subsets from R n (18)} be the MOD plane subset collecton. 1. Let Study questons () to (x) of problem (11) for ths B. a a a N = { a a a / a P(R n (1)); 1 1} 1. Let be the MOD plane matrx wth subset entres from R n (1). Study questons () to (x) of problem (11) for ths N wth approprate modfcatons. a W = { a a 1 a / a P(R n ()), 1 }

245 MOD Natural Neutrosophc Subset Topologcal be the MOD plane square matrx wth entres from subsets of P(R n ()). ) Study questons () to (x) of problem 11 for ths W wth approprate changes. ) {W, } = L be the MOD plane subset entres matrx semgroup. L s a noncommutatve semgroup. ) Prove W, W, W and W are the four n dstnct MOD plane subset entres matrx specal type of topologcal spaces whch are noncommutatve. v) Study relevant questons from () to (x) of problem (11) for these four MOD plane subset entres matrx spaces gven n (). 15. Let v) Obtan any other specal feature assocated wth four MOD specal types of topologcal spaces gven n queston (). a1 a a a a 5 a a 7 a 8 M = { / a P(R n (8)); 1 1} a9 a1 a11 a 1 a1 a1 a15 a1 be the MOD plane subset entres matrx collecton. ) Study questons () to (x) of problem (11) for ths M wth approprate changes.

246 MOD Subset Topologcal Spaces on the 5 ) ) Study questons () to (v) of problem (1) for ths M. Obtan any other specal feature enjoyed by ths M. 1. Let B = P(M) = {collecton of all subsets from a1 a M = { a / a R n (); 1 5}} a a 5 be the MOD plane matrx subset collecton. ) Prove {B, +}, {B, n }, {B, } and {B, } are MOD plane matrx subset semgroups. ) ) v) Prove all these four semgroups have MOD subsemgroups of both fnte and nfnte order. Whch of the MOD semgroups has deals of nfnte order? Prove {B, } has deals of fnte order. v) Can {B, } have deals of fnte or nfnte order? v) v) Fnd all MOD plane zero dvsors, MOD plane nlpotents and MOD plane dempotents n {B, n }. Prove {B, +} s only a MOD plane semgroup.

247 MOD Natural Neutrosophc Subset Topologcal v) What are the specal features enjoyed by B o = {B,, }, B = {B, +, n}, B n = {B, +, }, B n = {B, +, }, B and n B the sx MOD plane matrx subset topologcal spaces. Enumerate all the specal features assocated wth them. x) Can we say any of the MOD topologcal spaces satsfy the Kakutan s fxed pont theorem? x) Whch of the MOD topologcal spaces are compact? x) x) x) xv) Whch of the MOD topologcal spaces are connected? Prove B has MOD plane matrx subset specal type of topologcal subspaces whch are strong. Prove B cannot have MOD plane matrx subset specal type of topologcal space deals? Prove the exstence of MOD plane topologcal zero dvsors n B. xv) Obtan any other specal feature enjoyed by B. 17. Let S = {P(T)} = {collecton of all subsets from a1 a a T = { a a5 a a7 a8 a 9 / a R(); 1 9} be the MOD plane matrx subset collecton.

248 MOD Subset Topologcal Spaces on the 7 ) Study questons () to (xv) of problem (1) for ths S. ) ) Prove {S, } s a MOD plane matrx subset semgroup under the usual product. Prove {S, } s non commutatve and non assocatve MOD semgroup. v) Prove S, S, S and n S are all MOD plane matrx subset specal type of topologcal spaces whch are non commutatve. v) Can any of the four MOD plane matrx subset specal type of topologcal spaces satsfy the Kakutan s fxed pont theorem? v) v) Does the four spaces mentoned n queston () of the problem (a) connected? (b) compact? Obtan all specal features satsfed by these MOD plane non commutatve specal type of topologcal spaces. 18. Let W = P(V) = {collecton of all subsets from a a a a a a a a a a V = { a a a a a a a a a a a a a a a / a R n (8); 1 5} by the MOD plane matrx subset collecton.

249 8 MOD Natural Neutrosophc Subset Topologcal ) Study questons () to (xv) of problem (1) for ths W. ) Study questons () to (v) of problem (17) for ths W. 19. Let V = {P(B)} = {collecton of all matrx subsets from B = a a a a a a a a a a a a a a a a a a a a a / a R n (17), 1 1} be the MOD plane matrx subset collecton. Study questons () to (xv) of problem (1) for ths V.. Let T = S[x] = { ax / a P(R n (1))} be the MOD plane subset coeffcent polynomal collecton. ) Prove {T, +}, {T, }, {T, } and {T, } are MOD plane subset coeffcent polynomal semgroups. ) ) v) Prove T o, T, T, T, T and T are the four MOD plane subset coeffcent polynomal specal type of topologcal spaces. Do any one of them satsfy the Kakutan s fxed pont theorem? Prove {T, +} and {T, } does not contan nontrval MOD plane dempotents.

250 MOD Subset Topologcal Spaces on the 9 v) Are these MOD plane topologcal spaces () compact? () connected? v) v) Fnd MOD plane subset coeffcent polynomal specal type of strong topologcal subspaces. Prove T cannot have MOD plane subset coeffcent polynomal strong topologcal deals. v) Does T contan MOD plane subset coeffcent polynomal ¾ strong topologcal deals? x) Does T contan MOD plane subset coeffcent polynomal ½ strong topologcal deals? x) Characterze all MOD plane subset specal type of topologcal zero dvsors and nlpotents. x) Prove T cannot contan MOD plane subset specal type of topologcal dempotents n T. x) Obtan all specal features assocated wth T. 1. Let S = { a x / a P(R n (17))} be the MOD plane subset coeffcent polynomal collecton. Study questons () to (x) of problem () for ths S.. Let T = { a x / a P(R n (8))} be the MOD plane subset coeffcent polynomal collecton. Study questons () to (x) of problem () for ths T.. Let M ={P(S) where S = { a x MOD plane polynomal subset collecton. / a R n ()}} be the ) Study questons () to (x) of problem () for ths M.

251 5 MOD Natural Neutrosophc Subset Topologcal ) Enumerate all specal features assocated wth M.. Let T = {P(N) where N = { a x MOD plane polynomal subset collecton. / a R n ()}} be the ) Study questons () to (x) of problem () for ths T. ) Compare ths T wth M of problem (). 5. Let S = {P(T[M] 1 )) = { ax / a R n (), x 11 = 1}} 1 be the MOD plane fnte degree polynomal subset collecton. ) Study questons () to (x) of problem () for ths S. ) ) Dstngush ths wth problem n whch the degree of the polynomal s nfnte. Obtan any other specal and strkng features assocated wth ths.. Let W ={P(M[x] )} = {collecton of all subsets from M[x] = { a x / a R n (8); x = 1}} be the MOD plane fnte degree polynomal subsets. ) Study questons () to (x) of problem () for ths W ) Compare ths W wth S of problem (5).

252 MOD Subset Topologcal Spaces on the Let M = { 8 a x / a P(R n ()); x 9 = 1} be the MOD plane subset coeffcent fnte degree polynomals collecton. ) Study questons () to (x) of problem () for ths M. ) Compare ths M wth W of problem (). ) Obtan all specal features enjoyed by ths M. 8. Let B = { 1 a x / a P(R n (1)); x 1 = 1} be the MOD plane subset coeffcent fnte degree polynomal collecton. ) Study questons () to (x) of problem () for ths B. ) Compare ths B wth M of problem (7). 9. Let D = { 5 a x / a P(R n (9)); x = 1} be the MOD plane fnte degree polynomal subset coeffcent collecton. ) Study questons () to (x) of problem () for ths D. ) Compare ths D wth B of problem (8).. Let S = R = {a + b / a, b R, = } be the real neutrosophc plane. V = P(S) ={collecton of all subsets from S}; {V, +}, {V, }, {V, } and {V,} be the real

253 5 MOD Natural Neutrosophc Subset Topologcal plane neutrosophc semgroups. V o, V, V, V, V and V be the real neutrosophc topologcal spaces of specal type. t s left as an open conjecture whether any of these sx spaces satsfy the Kakutan s fxed pont theorem. 1. Let R g = {a + bg / a, b R, g = } be the dual number plane. Let B = P(R g) be the subsets collecton of dual numbers. Can any one of these dual number specal type of topologcal spaces, B o, B, B, B, B and B satsfy the Kakutan s fxed pont theorem.. Study problem (1) n whch R g s replaced by R h = {a + bh / a, b R, h = h}.. Study problem (1) n whch R g s replaced by R k = {a + bk / a, b R, k = k}.. Study Kakutan fxed pont theorem for the followng MOD nterval / plane specal type of topologcal spaces, A, B, C, D and E where; ) A = P([, n) g) = {collecton of subsets from the MOD nterval dual numbers}. A o, A, A, A, A and be the MOD nterval / plane dual number specal type of subset topologcal spaces. ) B = P([, n) h = {collecton of all subsets from the MOD nterval / plane specal dual lke numbers}. B o, B, B, B, B and B be the MOD nterval / plane specal dual lke number subset specal type of topologcal spaces.

254 MOD Subset Topologcal Spaces on the 5 ) C ={P([, n) k)} = {collecton of all subsets from [, n) k ={a + bk / a, b [, n), k = (n 1)k}} MOD nterval / plane specal quas dual number subsets collecton. C o, C, C, C, C and C be the MOD nterval / plane specal quas dual number subset specal type of topologcal spaces bult on C. v) D ={P([, n) )} ={collecton of all subsets from the MOD nterval / plane neutrosophc subsets from [, n) = {a + b / a, b [, n), = }} be the MOD nterval / plane natural neutrosophc subset collecton be the MOD nterval / plane natural neutrosophc subset specal type of topologcal spaces. v) E = {P(C([, n)))} = {collecton of all subsets from C([, n)) = {a + b F / a, b [, n), F = n 1 be the MOD plane / nterval fnte complex number collecton, E o, E, E, E, E and E be the MOD plane / nterval fnte complex number subset specal type of topologcal spaces. t s to be noted that ths problem can be vewed as an open conjecture on MOD specal type of topologcal spaces for the valdty of the Kakutan s fxed pont theorem. 5. Obtan all specal features enjoyed by MOD natural neutrosophc subset semgroups.. Let S = P( Z n ) = {Collecton of all subsets from Z n = {a + / a Z n and t Z n s a zero or a zero dvsor or an n t dempotent or a nlpotent}} be the MOD natural neutrosophc subset collecton.

255 5 MOD Natural Neutrosophc Subset Topologcal ) Prove {S, +}, {S, }, {S, } and {S, } are MOD natural neutrosophc subset semgroups of fnte order. ) ) v) Prove S o, S, S, S, S and S are MOD natural neutrosophc subset specal type of topologcal spaces of fnte order. Fnd all MOD natural neutrosophc subset topologcal zero dvsors, dempotents and nlpotents of the sx spaces mentoned n (). Are these spaces connected? v) Are these compact? v) v) v) x) Can the modfed form Kakutan fxed pont theorem be establshed for ths S? Prove these have MOD natural neutrosophc strong subspaces. Can these spaces have MOD strong deals? Obtan any other specal feature assocated wth ths S. 7. Let M = P ( Z ) = {collecton of subsets from the MOD 9 natural neutrosophc set Z 9 } be the MOD natural neutrosophc subset collecton. Study questons () to (x) of problem for ths M. 8. Let B = P(T) = {collecton of all subsets from a1 T ={ a a / a Z 1, 1 }} be the MOD natural

256 MOD Subset Topologcal Spaces on the 55 neutrosophc matrx subsets. B o, B, B, n B, n B n and B be the MOD natural neutrosophc matrx subset specal type of topologcal spaces. Can modfed or specal form of Kakutan s fxed pont theorem be defned and made vald for these spaces. mentoned above? 9. Let M = {collecton of all subset matrces a a a a a a 1 a a 5 a 7 8 a 9 1 /. Let a P( Z 8 ), 1 1} be the MOD natural neutrosophc matrx wth entres as subsets collecton. M o, M, M, M, n M and n be the MOD natural n M neutrosophc subset matrx specal type of topologcal spaces. Study queston (8) for ths M. a a a W = { a a a / a P(Z 1 g ; 1 18} be the MOD natural neutrosophc dual number subset matrx entres collecton. Let W o, W, W, W, n W n W n and be the MOD natural neutrosophc dual number subset matrx entres specal type of topologcal spaces. Study queston (8) for these topologcal spaces constructed usng W.

257 5 MOD Natural Neutrosophc Subset Topologcal 1. Let a1 a a M = { a a 5 a / a P(Z 8 ) 1 } be the MOD natural neutrosophc neutrosophc subset matrx entres collecton. M o, M, M, M, n M and n M be the MOD natural neutrosophc-neutrosophc matrx wth subset entres specal type of topologcal spaces.study queston (8) for these MOD specal topologcal spaces constructed usng M. n a1 a a. Let W = { a a5 a a7 a8 a 9 / a P(Z g ); 1 9} be the MOD natural neutrosophc dual number subset entres matrx collecton. W o, W, W, W, W, W, W, n W, n W and n W be MOD natural neutrosophc dual number subset entres matrx specal type of topologcal spaces. Study questons (8) for these 1 MOD topologcal spaces constructed usng W. n

258 MOD Subset Topologcal Spaces on the 57 a1 a a a a5 a a7 a 8. Let S = { a9 a1 a11 a 1 a1 a1 a15 a1 / a P(C (Z 1 ), 1 1} be the MOD natural neutrosophc fnte complex number subset matrx. Let S o, S, S, S n, n S, S, S, S, n S and n S be the MOD natural neutrosophc fnte complex number subset matrx specal type of topologcal spaces. Study queston (8) for these spaces.. Let A = { ax / a P( Z 1 )} be the MOD natural neutrosophc subset coeffcent polynomals collecton. A o, A, A, A, A, A and A be the MOD natural neutrosophc subset coeffcent polynomal specal type of topologcal spaces. Study queston (8) for these spaces. 5. Let B = {P(M[x])} = {collecton of all subsets from M[x] = { ax / a Z 7 }} be the MOD natural neutrosophc polynomal subset collecton. Let B o, B, B, B, B and B be the MOD natural neutrosophc polynomal subset specal type of topologcal spaces. Study queston (8) for these spaces bult usng B.

259 58 MOD Natural Neutrosophc Subset Topologcal. Let C = {P(N[x]) = {collecton of all subsets from N[x] = { a x / a Z }} be the MOD natural neutrosophc polynomal subsets collecton. C o, C, C, C, C and C be the MOD natural neutrosophc polynomal subsets specal type of topologcal spaces. Study queston (8) for these spaces bult usng C. 7. Let D ={P(M[x] 1 )} = {collecton of all subsets from 1 M[x] 1 = { ax / x 11 = 1, a Z 1 }} be the MOD natural neutrosophc fnte degree polynomal subset collecton. D o, D, D, D, D and D be the MOD natural neutrosophc fnte degree polynomal subset specal type of topologcal spaces. Study queston (8) for these spaces bult usng D Let S = { ax / a P( Z 8 ), x 9 = 1} be the MOD natural neutrosophc fnte degree subset coeffcent polynomal collecton. S o, S, S, S, S and S be the MOD natural neutrosophc subset coeffcent fnte degree polynomal specal type of topologcal spaces bult usng S. Study queston (8) for ths S Let H = { ax / a P( Z 19 ), x = 1} be the MOD natural neutrosophc subset coeffcent fnte degree polynomal collecton. H o, H, H, H, H and H be the MOD natural neutrosophc subset coeffcent

260 MOD Subset Topologcal Spaces on the 59 fnte degree polynomal specal type of topologcal spaces bult usng H. Study queston (8) for ths H. 5. Let K = { ax / a P(C (Z 1 ))} be the MOD natural neutrosophc fnte degree complex number subset coeffcent polynomal collecton. K o, K, K, K, K and K be the MOD natural neutrosophc fnte complex number subset coeffcent polynomal specal type of topologcal spaces. Study queston (8) for ths K. 51. Study for problem (5) when P(C (Z 1 )) s replaced by P(Z 17 g ), P(Z h ), P(Z k ) and P(Z 1 ) and analyse queston (8) and dstngush the specal features assocated wth them. 5. Let L = {P(M[x])} ={collecton of all subset polynomals from M[x] = { ax / a Z 1 }} be the MOD natural neutrosophc-neutrosophc polynomal subset collecton. L o, L, L, L, L and L be the MOD natural neutrosophc-neutrosophc polynomal subset specal type of topologcals spaces. Study questons (8) for ths L. 5. Study L n problem (5) when Z 1 s replaced by Z 8 g, Z 7 h, Z 1 k, and C (Z 9 ), and analyse the queston (8) for these spaces. Dstngush the specal features enjoyed by these spaces.

261 MOD Natural Neutrosophc Subset Topologcal 8 5. Let M = { ax / a P(Z 19 ); x 9 = 1} be the MOD natural neutrosophc-neutrosophc fnte degree subset coeffcent polynomal collecton. M o, M, M, M, M and M be the MOD natural neutrosophcneutrosophc fnte degree polynomals wth subset coeffcent specal type of topologcal spaces. Study queston (8) for ths M. 55. Study queston (5) f Z 19 s replaced by Z 18 g, Z g, Z 8 h, Z 119 g and C (Z ), the queston (8) for the respectve topologcal spaces. 5. Let P ={P(M[x] 8 )} = {collecton of all subsets from 8 M[x] 8 = { ax / x 9 = 1, a C (Z 8 be the MOD natural neutrosophc fnte complex number fnte degree polynomal subset collecton from M[x] 8. P o, P, P, P, P and P be the MOD natural neutrosophc fnte complex number fnte degree polynomal subset specal type of topologcal spaces. Study queston (8) for ths P. 57. Replace n problem (5) C (Z 8 ) by Z g Z h, Z 7 k and Z 1 and study problem (8) for these mod topologcal spaces. 58. Let M = {{P( [, 8)) = {collecton of all subsets from 8 [, 8) = {a + t / a [, 8), t s a zero dvsor or dempotent or nlpotent n Z 8 }} be the MOD natural neutrosophc subset collecton. Let S o, S, S, S, S and S be the MOD natural neutrosophc subset specal

262 MOD Subset Topologcal Spaces on the 1 type of topologcal space. Study problem (8) for all the spaces bult usng S. 59. Let T = {P([, ) g )} = {collecton of all subsets from [, ) g = {a + bg + t / a, b [, ), g = t s an dempotent or nlpotent or zero dvsor n Z }} be the MOD natural neutrosophc dual number subset collecton. T o, T, T, T, T and T be the MOD natural neutrosophc dual number subset specal type of topologcal space usng T. Study queston (8) for ths T.. f n (59) when [, ) g s replaced by C ([,1), [, 17) h, [, 11) k and [,1) for the respectve MOD specal nterval topologcal spaces study queston (8). a1 a 1. Let M = { / a P( [, )); 1 1} be the a1 MOD nterval natural neutrosophc matrx wth subset entres collecton. M o, M, M M, n M, and n n M be the MOD nterval natural neutrosophc matrx wth subset entres specal type of topologcal spaces bult usng M. Study queston (8) for ths M.. f P( [, )) n problem 1 s replaced by P([, 7) g, P([, 19) k ), P ([, 8) ) and P([,9) h ) for ther respectve topologcal spaces. Study queston (8).

263 MOD Natural Neutrosophc Subset Topologcal. Let W = {P(M)} = {collecton of all subsets from a1 a a a M = { a5 a a 7 a 8 where a [, 8) g ; 1 8}} be the MOD nterval natural neutrosophc dual number matrx subset collecton. W o, W, W W, n W, and n be the n W MOD nterval natural neutrosophc dual number matrx subset specal type of topologcal spaces assocated wth W. Study queston (8) for these MOD specal type of topologcal spaces assocated wth W.. n problem replace the space W by [, 8) g ths by C ([, 9)), [,8), [,19) h, [, ) k. Study problem 8 for all these above mentoned spaces. 5. Let N = { ax / a P( [, n)) be the MOD nterval natural neutrosophc subset coeffcent polynomal collecton. N o, N, N, N, N and N be the MOD nterval natural neutrosophc subset coeffcent polynomal specal type of topologcal spaces. Study queston (8) for ths N.. Replace n problem 5, P( [, n)) by P([, 18) g ), P([,19) h, P([, 8) k ), P([, ) and study queston (8) for all these MOD topologcal space. ) Dstngush ther smlartes and dssmlartes. ) Compare them wth each other.

264 MOD Subset Topologcal Spaces on the 7. Let V = P(S[x]) = {collecton of all subsets from S[x] = { ax / a [, 7) be the MOD natural neutrosophc polynomal subset collecton. Study queston (8) for ths V. 8. n problem (7) replace [,7) by [, ) g, [, 8) h, [, 19) k, [, ) and C ([, 7)) and study the queston (8) for these respectve sets MOD specal type of topologcal spaces. 9. Let Z = { ax / a P( [, 1)) be the MOD nterval natural neutrosophc subset coeffcent polynomal collecton. Z o, Z, Z, Z, Z and Z be the MOD nterval natural neutrosophc subset coeffcent polynomal specal type of topologcal spaces. Study queston (8) for ths Z. 7. Replace n problem 9, P( [, 1)) by P([, 7) g ), P([,8) ), P([, 11) h ), C ([, )) and P([, ) k ). Study the topologcal spaces assocated wth these sets the queston (8) Let Y = { ax / a P( [, )); x 19 = 1 be the MOD nterval natural neutrosophc fnte degree subset coeffcent polynomal collecton. Let Y o, Y, Y, Y, Y and Y be the MOD nterval natural neutrosophc fnte degree subset coeffcent polynomal specal type of topologcal spaces.

265 MOD Natural Neutrosophc Subset Topologcal Study queston (8) for the all types of sets of topologcal spaces bult usng Y. 7. Replace n problem 71, P( [, )) n Y of problem (71) by P(C [,8)), P([, 17) ), P([, 81) g ), P([, 1) k ) and P([, ) h ) and study for those MOD specal topologcal spaces the problem Let B = {P(M[x] )} = {collecton of all subsets from M[x] = { ax / x 1 = 1, a [, )}} be the MOD nterval natural neutrosophc fnte degree polynomal subset collecton. B o, B, B, B, B and B be the MOD nterval natural neutrosophc fnte degree polynomal specal type of topologcal spaces. Study queston (8) for ths B. 7. Replace B of problem (7) the set [, ) by C ([,5)); [, 9) g, [, 1) h, [, 11) k and [, ) and for the related 5 dstnct MOD nterval topologcal spaces and study queston (8).

266 FURTHER READNG 1. Kakutan, S., Selected Papers, Robert R. Kallman (edtor), Smarandache, F. Neutrosophc Logc - Generalzaton of the ntutonstc Fuzzy Logc, presented at the Specal Sesson on ntutonstc Fuzzy Sets and Related Concepts, of nternatonal EUSFLAT Conference, Zttau, Germany, 1-1 September. Smarandache, F., Collected Papers, Edtura Abaddaba, Oradea,. Vasantha Kandasamy, W.B. and Smarandache, F., nterval Semgroups, Kappa and Omega, Glendale, (11). 5. Vasantha Kandasamy, W.B. and Smarandache, F., Fnte Neutrosophc Complex Numbers, Zp Publshng, Oho, (11).. Vasantha Kandasamy, W.B. and Smarandache, F., Dual Numbers, Zp Publshng, Oho, (1). 7. Vasantha Kandasamy, W.B. and Smarandache, F., Specal dual lke numbers and lattces, Zp Publshng, Oho, (1).

267 MOD Natural Neutrosophc Semrngs 8. Vasantha Kandasamy, W.B. and Smarandache, F., Specal quas dual numbers and Groupods, Zp Publshng, Oho, (1). 9. Vasantha Kandasamy, W.B., and Smarandache, F., Set deal Topologcal Spaces, Zp Publshng, (1). 1. Vasantha Kandasamy, W.B., and Smarandache, F., Quas Set Topologcal vector subspaces, Zp publshng, (1). 11. Vasantha Kandasamy, W.B. and Smarandache, F., Algebrac Structures usng Subsets, Educatonal Publsher nc, Oho, (1). 1. Vasantha Kandasamy, W.B. and Smarandache, F., Algebrac Structures usng [, n), Educatonal Publsher nc, Oho, (1). 1. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., Specal type of subset topologcal spaces, Educatonal Publsher, (1). 1. Vasantha Kandasamy, W.B., and Smarandache, F., Subset non Assocatve Topologcal Spaces, Educatonal Publsher, (1). 15. Vasantha Kandasamy, W.B., and Smarandache, F., Pseudo Lattce graphs and ther applcatons to fuzzy and neutrosophc models, EuropaNova, (1). 1. Vasantha Kandasamy, W.B. and Smarandache, F., Algebrac Structures on the fuzzy nterval [, 1), Educatonal Publsher nc, Oho, (1). 17. Vasantha Kandasamy, W.B. and Smarandache, F., Algebrac Structures on Fuzzy Unt squares and Neutrosophc unt square, Educatonal Publsher nc, Oho, (1). 18. Vasantha Kandasamy, W.B. and Smarandache, F., Natural Product on Matrces, Zp Publshng nc, Oho, (1).

268 Further Readng Vasantha Kandasamy, W.B. and Smarandache, F., Algebrac Structures on Real and Neutrosophc square, Educatonal Publsher nc, Oho, (1).. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., MOD planes, EuropaNova, (15). 1. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., MOD Functons, EuropaNova, (15).. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., Multdmensonal MOD planes, EuropaNova, (15).. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., Natural Neutrosophc numbers and MOD Neutrosophc numbers, EuropaNova, (15).. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., Algebrac Structures on MOD planes, EuropaNova, (15). 5. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., MOD Pseudo Lnear Algebras, EuropaNova, (15).. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., Problems n MOD Structures, EuropaNova, (1). 7. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., Semgroups on MOD Natural Neutrosophc Elements, EuropaNova, (1). 8. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., Specal Type of Fxed Ponts of MOD Matrx Operators, EuropaNova, (1). 9. Vasantha Kandasamy, W.B., lanthenral, K., and Smarandache, F., Specal Type of Fxed Ponts pars of MOD Rectangular Matrx Operators, EuropaNova, (1).

269 8 MOD Natural Neutrosophc Semrngs. Vasantha Kandasamy, W.B., Smarandache, F. and lanthenral, K., MOD Cogntve Maps models and MOD natural neutrosophc Cogntve Maps models, EuropaNova, (1). 1. Vasantha Kandasamy, W.B., Smarandache, F. and lanthenral, K., MOD Relatonal Maps models and MOD natural neutrosophc relatonal Maps models, EuropaNova, (1).. Vasantha Kandasamy, W.B., Smarandache, F. and lanthenral, K., MOD Natural neutrosophc subset semgroups, EuropaNova, (1).

270 NDEX D Dual number plane, - K Kakutan s conjectures, -8 Kakutan s fxed pont theorem, 1- M MOD dual number plane, - MOD fnte complex number nterval subset topologcal deals, 11- MOD fnte complex number nterval subset topologcal spaces, 11- MOD dempotent semgroups, 1- MOD nterval dual number semgroups, 111- MOD nterval fnte complex number subset semgroups, 11-

271 7 MOD Natural Neutrosophc Subset Topologcal MOD nterval fnte dual number subset semgroups, 11-5 MOD nterval natural neutrosophc specal quas, dual number subset matrx semgroup, 15- MOD nterval natural neutrosophc dual number subset coeffcent polynomal specal type of topologcal spaces, 17-9 MOD nterval natural neutrosophc fnte complex number subset coeffcent polynomal semgroup, 1-8 MOD nterval natural neutrosophc fnte complex number subset coeffcent polynomal specal type of topologcal spaces, 1-9 MOD nterval natural neutrosophc fnte complex number subset coeffcent polynomal specal type of topologcal subspaces, 1-9 MOD nterval natural neutrosophc specal dual lke number subset coeffcent polynomal specal type of topologcal spaces, 15- MOD nterval natural neutrosophc specal dual lke number subset semgroup, 11-7 MOD nterval natural neutrosophc specal dual lke number subset coeffcent polynomal semgroups, 15- MOD nterval natural neutrosophc specal quas dual number matrx subset specal type of topologcal spaces, MOD nterval natural neutrosophc specal quas dual number subset matrx topologcal spaces, 1-

272 MOD nterval natural neutrosophc specal quas dual number subset coeffcent polynomal semgroups, MOD nterval natural neutrosophc specal quas dual number subset coeffcent polynomal specal type of topologcal spaces, MOD nterval natural neutrosophc subset coeffcent polynomal semgroups, 15-7 MOD nterval natural neutrosophc subset coeffcent polynomal specal type of topologcal spaces, 15-7 MOD nterval natural neutrosophc-neutrosophc subset matrx semgroup, 11-9 MOD nterval natural neutrosophc-neutrosophc subset matrx topologcal space, 11-9 MOD nterval specal dual lke number subset coeffcent polynomal specal type of topologcal spaces, MOD nterval specal dual lke number subset coeffcent polynomal semgroups, MOD nterval subset coeffcent polynomal semgroup, 11- MOD nterval subset coeffcent polynomal specal type of topologcal subspaces, 1-5 MOD nterval subset coeffcent polynomal subset specal type of topologcal spaces, 1-5 MOD nterval subset matrx specal type of topologcal spaces, 11-7 MOD nterval subset semgroup, MOD nterval subset specal type of topologcal space of nlpotents, 111- ndex 71

273 7 MOD Natural Neutrosophc Subset Topologcal MOD nterval subset specal type of topologcal subspaces, 11- MOD nterval subset specal type of topologcal zero dvsors, 11- MOD nterval subset specal type topologcal space dempotents, 11- MOD nterval subset strong topologcal subspaces, 11-5 MOD nterval subset subsemgroups, 11- MOD natural neutrosohc-neutrosophc specal type of subset topologcal spaces, - MOD natural neutrosohpc specal type of ¾ strong subsemgroups, subset topologcal spaces 7-9 MOD natural neutrosophc dual number specal type of subset topologcal spaces, -5 MOD natural neutrosophc dual number specal type of subset topologcal subspaces, -5 MOD natural neutrosophc dual number subset semgroup, -5 MOD natural neutrosophc fnte complex number subsets specal type of topologcal space, -8 MOD natural neutrosophc nterval dual number subset specal type of topologcal spaces, 11- MOD natural neutrosophc specal dual lke number subset semgroups, -8 MOD natural neutrosophc specal dual lke number subset topologcal spaces, -8 MOD natural neutrosophc specal dual lke number subset topologcal subspaces, -8 MOD natural neutrosophc specal quas dual number subset semgroups, 7-8

274 ndex 7 MOD natural neutrosophc specal quas dual number subset specal type of topologcal spaces, 7-8 MOD natural neutrosophc specal quas dual number subset specal type of topologcal subspaces, 7-9 MOD natural neutrosophc specal type of ½ strong subsemgroup subset topologcal spaces, 1-5 MOD natural neutrosophc specal type of fnte complex number subset topologcal subspaces, 7-9 MOD natural neutrosophc specal type of subset topologcal spaces subsemgroup, -8 MOD natural neutrosophc specal type of subset topologcal space strong subsemgroup, -9 MOD natural neutrosophc specal type of subset topologcal space strong subsemgroup, -9 MOD natural neutrosophc specal type of topologcal spaces, 1-8 MOD natural neutrosophc specal type of topologcal subset space deal, 5-7 MOD natural neutrosophc subset matrx specal type of semgroups, -5 MOD natural neutrosophc subset matrx specal type of topologcal spaces, -5 MOD natural neutrosophc subset semgroups, 1-9 MOD natural neutrosophc subset subsemgroups, -7 MOD natural neutrosophc-neutrosophc specal type of subset topologcal subspaces, - MOD natural neutrosophc-neutrosophc specal type of subset topologcal subspaces assocated wth strong subset semgroups, 1-.

275 7 MOD Natural Neutrosophc Subset Topologcal MOD plane subset specal type of topologcal spaces, 1-5 MOD semgroup, 9-1 MOD semlattce, 9-1 MOD subsemgroup completon, 1-1 MOD subsemgroups, 1-1 MOD subset dual number polynomal semgroup, 7-9 MOD subset nterval specal type of topologcal spaces, 11- MOD subset matrx semgroups, 5- MOD subset matrx specal type of non-commutatve topologcal spaces, 5- MOD subset matrx specal type of topologcal dempotents, - MOD subset matrx specal type of topologcal nlpotents, - MOD subset matrx specal type of topologcal spaces, 5-5 MOD subset matrx specal type of topologcal zero dvsors, - MOD subset natural neutrosophc polynomal specal type of topologcal spaces, 7-9 MOD subset natural neutrosophc-neutrosophc specal type of polynomal topologcal spaces, 9-7 MOD subset polynomal specal type of topologcal spaces, 7-9 MOD subset topologcal spaces of specal type, 15-8 MOD topologcal strong subset subspaces, 1-5

276 ndex 75 N Neutrosophc plane, 9-1 R Real specal quas dual number plane, -9 S Specal type of MOD subset topologcal spaces, 15-8 T Type subset semrng, 15-7 Type subset semrngs, 15-7

277 ABOUT THE AUTHORS Dr.W.B.Vasantha Kandasamy s a Professor n the Department of Mathematcs, ndan nsttute of Technology Madras, Chenna. n the past decade she has guded 1 Ph.D. scholars n the dfferent felds of non-assocatve algebras, algebrac codng theory, transportaton theory, fuzzy groups, and applcatons of fuzzy theory of the problems faced n chemcal ndustres and cement ndustres. She has to her credt 9 research papers. She has guded over 1 M.Sc. and M.Tech. projects. She has worked n collaboraton projects wth the ndan Space Research Organzaton and wth the Taml Nadu State ADS Control Socety. She s presently workng on a research project funded by the Board of Research n Nuclear Scences, Government of nda. Ths s her 11 st book. On nda's th ndependence Day, Dr.Vasantha was conferred the Kalpana Chawla Award for Courage and Darng Enterprse by the State Government of Taml Nadu n recognton of her sustaned fght for socal justce n the ndan nsttute of Technology (T) Madras and for her contrbuton to mathematcs. The award, nsttuted n the memory of ndan-amercan astronaut Kalpana Chawla who ded aboard Space Shuttle Columba, carred a cash prze of fve lakh rupees (the hghest prze-money for any ndan award) and a gold medal. She can be contacted at vasanthakandasamy@gmal.com Web Ste: or Dr. K. lanthenral s Assstant Professor n the School of Computer Scence and Engg, VT Unversty, nda. She can be contacted at lanthenral@gmal.com Dr. Florentn Smarandache s a Professor of Mathematcs at the Unversty of New Mexco n USA. He publshed over 75 books and artcles and notes n mathematcs, physcs, phlosophy, psychology, rebus, lterature. n mathematcs hs research s n number theory, non-eucldean geometry, synthetc geometry, algebrac structures, statstcs, neutrosophc logc and set (generalzatons of fuzzy logc and set respectvely), neutrosophc probablty (generalzaton of classcal and mprecse probablty). Also, small contrbutons to nuclear and partcle physcs, nformaton fuson, neutrosophy (a generalzaton of dalectcs), law of sensatons and stmul, etc. He got the 1 Teleso-Galle Academy of Scence Gold Medal, Adjunct Professor (equvalent to Doctor Honors Causa) of Bejng Jaotong Unversty n 11, and 11 Romanan Academy Award for Techncal Scence (the hghest n the country). Dr. W. B. Vasantha Kandasamy and Dr. Florentn Smarandache got the 1 New Mexco-Arzona and 11 New Mexco Book Award for Algebrac Structures. He can be contacted at smarand@unm.edu

278