Multidimensional. MOD Planes. W. B. Vasantha Kandasamy Ilanthenral K Florentin Smarandache

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2 Multidimensionl MOD Plnes W. B. Vsnth Kndsmy Ilnthenrl K Florentin Smrndche 2015

3 This book cn be ordered from: EuropNov ASBL Clos du Prnsse, 3E 1000, Bruxelles Belgium E-mil: URL: Copyright 2014 by EuropNov ASBL nd the Authors Peer reviewers: Prof. Xingsen Li, School of Mngement, Ningbo Institute of Technology, Zhejing University, Ningbo , P. R. Chin Sid Broumi, University of Hssn II Mohmmedi, Hy El Brk Ben M'sik, Csblnc B. P. 7951, Morocco Prof. Vleri Kroumov, Dept. of Electricl nd Electronic Engineering, Fculty of Engineering, Okym University of Science, 1-1, Ridi-cho, Kit-ku, Okym Florentin Popescu, Fcultte de Mecnic, University of Criov, Romni. Mny books cn be downloded from the following Digitl Librry of Science: ISBN-13: EAN: Printed in the United Sttes of Americ 2

4 CONTENTS Prefce 5 Chpter One RENAMING OF CERTAIN MATHENMATICAL STRUCTURES AS MOD STRUCTURES 7 Chpter Two PROPERTIES OF MOD INTERVAL TRANSFORMATION OF VARYING TYPES 91 3

5 Chpter Three INTRODUCTION TO MOD PLANES 113 Chpter Four HIGHER AND MIXED MULTIDIMENSIONAL MOD PLANES 143 Chpter Five SUGGESTED PROBLEMS 187 FURTHER READING 224 INDEX 227 ABOUT THE AUTHORS 232 4

6 PREFACE In this book uthors nme the intervl [0, m); 2 m s mod intervl. We hve studied severl properties bout them but only here on wrds in this book nd forthcoming books the intervl [0, m) will be termed s the mod rel intervl, [0, m)i s mod neutrosophic intervl, [0,m)g; g 2 = 0 s mod dul number intervl, [0, m)h; h 2 = h s mod specil dul like number intervl nd [0, m)k, k 2 = (m 1) k s mod specil qusi dul number intervl. However there is only one rel intervl (, ) but there re infinitely mny mod rel intervls [0, m); 2 m. The mod complex modulo finite integer intervl (0, m) i F ; i 2 F = (m 1) does not stisfy ny nice properly s tht intervl is not closed under product. Here we define mod trnsformtions nd discuss severl interesting fetures bout them. So chpter one of this book serves the purpose of only reclling these properties. Vrious 5

7 properties of mod intervl trnsformtion of vrying types re discussed. Next the mod plnes re introduced minly to mke this book self contined one. Finlly the new notion of higher nd mixed multidimensionl mod plnes re introduced nd their properties nlysed. Severl problems re discussed some of which re open conjectures. We wish to cknowledge Dr. K Kndsmy for his sustined support nd encourgement in the writing of this book. W.B.VASANTHA KANDASAMY ILANTHENRAL K FLORENTIN SMARANDACHE 6

8 Chpter One RENAMING OF CERTAIN MATHEMATICAL STRUCTURES AS MOD STRUCTURES The min im of this book is to define smll sets which more or less depicts the rel line or nturl number line or ny other s MOD. So by MOD mthemticl structure we men they re reltively very smll in comprison with the existing one. When these structures somewht represent the lrger one we cll them s semimod structures or deficit MOD structures or demi MOD structures. We will proceed onto describe nd develop them. We know Z n the set of modulo integer, 2 n < hs severl nice lgebric structures. Z n s re defined s the semimod pseudo intervls. The mp from η z : Z Z n is such tht η z (1) = 1, η z (2) = 2, η z (n 1) = n 1, η z (n) = 0 nd so on.

9 8 Multidimensionl MOD Plnes Thus η z (m) = m if 0 m n 1 nd if m = n then η z (m) = 0 if m > n then m n = t + r n then η z(m) = r. Thus we hve infinite number of elements in Z mpped on to single point. Further η z ( t) = m t if t < m nd if t > m then t m = s r m so tht η z(t) = m r. We will first illustrte this sitution by n exmple or two. Exmple 1.1: Let Z 7 be the ring of modulo integers. η z : Z Z 7 be MOD trnsformtion η z (x) = x if x {0, 1, 2,, 6}; η z (9) = 2 = η z (7 + 2) = s η z (7) = 0. η z ( 3) = 4, η z ( 15) = 6 nd so on. We cll the mp η z of the semi pseudo MOD intervl mp or semi MOD pseudo intervl or demi MOD pseudo intervl mp or deficit MOD pseudo intervl mp. η z is semi pseudo MOD intervl trnsformtion. η z : Z Z 15 be demi pseudo MOD intervl trnsformtion. η z (m) = m if; 0 m 15. η z (t) = η(15n + s) = s where 1 s < if t is negtive, η z (t) = η(15n s) = 15 s.

10 Renming of Certin Mthemticl Structures 9 Now Z n is clled the demi MOD pseudo intervl. η z is the demi MOD pseudo intervl trnsformtion. Clerly we hve T z : Z n Z is such tht T z is pseudo mpping. T z (x) = nt + x; 0 t <. T z is not even mp under the stndrd definition of the mp or function from the domin spce Z n to rnge spce Z. Hence T z is MOD pseudo trnsformtion. This is the first time we mp one to mny elements. Infct one cn define or mke Z into n equivlence clsses nd sy mp 0 t n into the corresponding equivlence clsses of Z but not into Z; or one hs to divide Z into equivlence clsses nd then define T z : Z n Z. Thus this is not usul function/mp. Such functions or mps re in bundnt. This will be proved soon in the following chpters. We cll Z n the MOD dul specil pseudo intervl. Infct (Z n, +) is the semi MOD group nd {Z n, } is the semi MOD semigroup. Finlly {Z n, +, } is semi MOD ring. {Z n, +, } is semi MOD field if n = p is prime. Here we re not interested to study in this direction. For the limittions of this book is to introduce certin MOD notions of intervls, plnes nd spces.

11 10 Multidimensionl MOD Plnes However the term MOD is used to represent the study is mde on minimized or reduced or mde smll the existing big structures with mximum cre tken to see tht the properties of the big structure re not chnged to lrge extent. Now η z : Z Z n is MOD trnsformtion of integers to the semi MOD pseudo structures (intervl). We see for some ske of uniformity nd to derive sort of link we define these well known structures lso s MOD structure for they re derived from the infinite line or the integer set Z. While trying to get from Q this MOD collection we cll it s MOD intervl. For we shll denote the MOD rtionl intervl by [0, m) q where the suffix q distinguishes it from the semi open intervl [0, m) of R rels defined in chpter II of this book. η q : Q [0, m) q ; MOD trnsformtion of rtionls to semi MOD pseudo rtionl intervl does not exist or cnnot be defined s s/t where t divides m is not defined in [0, m) q. Let us for concreteness tke [0, 9) q nd let η q : Q [0, 9) q (Recll [0, m) q contins ll rtionls lying between 0 nd 9 of course 9 is not included). For if x = Q then η q 5 = 24 5 [0, 9) q η 608 q 7 = 22 7 [0, 9) q 45 η q 13 = [0, 9) q

12 Renming of Certin Mthemticl Structures η q 17 = [0, 9) 5 q. η q is not defined. 9 This is the wy the pseudo semi MOD rtionl trnsformtion tkes plce but is not well defined s it is only selective. Let η q : Q [0, 12) q [0, 12) q = {ll elements from the rtionl between 0 nd 12; 12 not included in [0, 12) q }. Let 254 Q η q 3 = 2 3 [0, 12) q. η q ( 483) = 9 [0, 12) q. 427 η q 11 = 9 11 [0, 12) q. Thus we lso hve MOD rtionl intervls. We pln to work with MOD rel intervls in chpter II of this book. Now Z m i F = {i F Z m }; i 2 F = m 1. We cll Z m i F = {i F Z m } nd is defined s the MOD integer semi complex number. We will illustrte this by some exmples. Exmple 1.2: Let Z 10 i F = {i F Z 10 } be the MOD integer semi complex number. Thus Z 10 i F = {i F, 2i F, 3i F, 4i F, 5i F, 6i F, 7i F, 8i F, 9i F, 0}.

13 12 Multidimensionl MOD Plnes We see Z 10 i F is n belin group under +. {Z 10 i F, } is not even closed. So {Z 10 i F, +} is n belin group. We cn mp from Zi = {i Z}; i 2 = 1 into Z 10 i F, i = 9. 2 F We cn hve MOD pseudo mp from Zi to Z 10 i F in the following wy. η zi : Zi Z 10 i F ; η zi (49i) = 9i F first η zi (i) = i F, η zi ( 93i) = 7i F, η zi (28i) = i F, η zi (90) = 0, η zi ( 90) = 0. Let us tke the semi MOD complex modulo integer intervl Z m i F. Consider η zi : Zi Z 26 i F ; we show how the MOD demi integer complex intervl trnsformtion tkes plce. η zi (167i) = 11i F, η zi ( 427i) = 15i F, η zi (22i) = 22i, η zi ( 16i) = 10i F nd η zi (178i) = 22i F. Let η zi : Zi Z 13 i F is defined s follows. η(12i) = 12i F, η(27i) = i F, η( 12i) = i F, η( 25i) = i F. Thus we see 27i, 12i nd 25i Zi re mpped on to the sme element in Z 13 i F. Infct we hve infinitely mny elements in Zi which re mpped on to single element of Z 13 i F. This is true for ll Z m i F.

14 Renming of Certin Mthemticl Structures 13 We will show by some more exmples. Let us consider Z 15 i F. η zi : Zi Z 15 i F is such tht η zi (n 15i F ) = 0 for ll n Z +. η zi (16i F ) = η zi (31i F ) = η zi (46i F ) = η zi (65i F ) = η zi (76i F ) = η zi (91i F ) = = i F. In generl η zi (n15i F + i F ) = i F for ll n Z +. Thus infinitely mny points in Zi re mpped on to single point in Z m i F. Infct we cn sy ll those points which re mpped to single point re periodic. This is shown in the bove exmple. η zi (17i) = η zi (2i) = η zi (32i) = η zi (47i) = = η zi (n15i + 2i) = 2i F for n Z +. Another nturl question would be wht hppens when elements in Zi re negtive. We will illustrte the sitution before we generlize. η zi ( 14i) = i F, η zi ( 29i) = i F, η zi ( 44i) = i F nd so on. η zi ( 15ni 14i) = i F for n Z +. Thus ll these elements re periodic nd re mpped on to the single element i F. Consider η zi ( 13i) = 2i F = η zi ( 28i) = η zi ( 43i) nd so on.

15 14 Multidimensionl MOD Plnes Thus for negtive vlues in Zi is n infinite number of periodic elements nd re mpped on to single element in Z n i F. Hence the mpping is nice mpping done in regulr periodic nd in systemtic wy. Now hving seen the properties enjoyed by Z n i F nd its reltion to Zi the integer imginry line we now proceed on to study cn we get bck Zi given Z n i F in unique wy. The nswer is yes. First this is illustrted by some exmples. Exmple 1.3: Let Z 7 i F be the complex modulo integers {0, i F, 2i F, 3i F,, 6i F }. Cn we get bck Zi? The nswer is yes. For ll elements mpped in this wy by η z 7iF We shll denote them by [0], [i F ], [2i F ],, [6i F ]. η z 7 i F ([0]) = {0, n7i, n Z}. η ([i F ) = {0, n7i + i, n 1 7i 6i, n 1 Z, n Z + }. z 7 i F z 7 i F : Z 7 i F Z re. η ([2i F ]) = {n7i + 2i, 0, n 1 7i 5i, n 1 Z, n Z + } likewise for other terms in Z 7 i F. Thus the mp is visulized in different wy further this is not function in the usul sense. This wy of mpping is minly done for in cse of [0, m) q i F ; x [0, m) q re only rtionls is subcollection from Q nd [0, m) q i F ; x [0, m) q nd x R re to be defined nd they cnnot be conveniently put into this form s in cse of Z m i F.

16 Renming of Certin Mthemticl Structures 15 Now hving seen specil pseudo trnsformtion which by ll mens is not function in the usul form. We now proceed on to define functions η q i F : Q [0, m) q i F nd its corresponding pseudo mps which re not usul functions lso known s MOD intervl trnsformtion nd pseudo intervl trnsformtion respectively. Exmple 1.4: Let [0, 9) q i F be the MOD rtionl complex modulo integer intervl with i 2 F = 8. Now η q i F η q i (5i) = 5i F. F η q if 426i 5 : Qi [0, 9) q i F ; 5i Qi is such tht η q i (i) = i F. F = i F nd so on. η q i (7/8i) = 7/8i F. F Clerly the intervl [0, m) q i F contins ll rtionls of the form p/q; q 0 nd p nd q integers q < m. Let η q if 478i 7 = 3 5 if. 7 η q if 7i 13 = 6 13 i F nd so on. But for i Qi is not defined in [0, 9) qi F. All elements ti/s where s = 3 or 6 is lso not defined in [0, 9) q i F. Thus, this wy of opertion of MOD trnsformtion from Qi to [0, m) q i F is not well extended.

17 16 Multidimensionl MOD Plnes Exmple 1.5: Let [0, 18) q i F be the MOD complex finite modulo integer intervl. η q i F trnsformtion. : Qi [0, 18) q i F is the MOD complex intervl η q i (i) = i F, F η q i (5i) = 5i F, F η q i (486i) = 0, F η q i ( 10i) = 8i F, F η q i ( 216i) = 0, F divides 18 nd so on. η q i ( 219i) = 15i F, F η q if 49 i 2 is not defined s 2 Infct the MOD rtionl complex modulo integer 1 trnsformtion is not possible s i hs no element in 18 [0, 18) q i F or 1 i where t-divides 18. t Exmple 1.6: Let [0, 11) q i F be the MOD rtionl complex 3 number intervl η q if i 4 = 3 if, η q i (i) = i F, 4 F η q if η q if 483 i i 4 = i F = i F nd so on. Now wht is the lgebric structure enjoyed by [0, m) q i F?

18 Renming of Certin Mthemticl Structures 17 Clerly [0, m) q i F is group under + nd it is not even closed under the product opertion. We will illustrte this sitution by n exmple or two. Exmple 1.7: G = [0, 6) q i F is MOD complex rtionl number group. 0 G nd 0 + α = α + 0 for ll α G. Let x = 0.85i F G then y = 5.15i F G is such tht y + x = x + y = 0. Let x = 92i F 100 nd y = 7iF 10 G s (100, 6) 1 nd (10, 6) 1. In generl 1 1 or is not even defined in G. However for ll prcticlities we ssume [0, m) q i F contins p/q q 0, 0 < q < m, p nd q integers nd if pi nd qi Qi yet pi qi = pq is not in Qi so product is not defined even in Qi. Thus if x = i Qi then we know η i(x) is in [0, 6) q i F. Hence our study revels the intervl [0, m) q or [0, m) q i F hs no proper trnsformtion. So t this juncture it is left s open conjecture how [0, m) q or [0, m) q i F cn hve t lest prtilly defined MOD trnsformtion. Thus from now onwrds we cn hve MOD intervls [0, m) [0, ); only rels nd we do not in this book venture to study [0, m) q or [0, m) q i F.

19 18 Multidimensionl MOD Plnes However this study is left open for ny resercher. Let us consider [0, m), m <, x [0, m) re rels from the rel line [0, m). Then we cn mp R [0, m) r in nice wy. For ll elements hve deciml representtion so one need not go for the concept p/q; q 0, q m. p/q = some rel deciml, here [0, m) r is lso denoted by [0, m). For 1/3 in R is nd is defined in [0, 3) wheres 1/3 is not defined in [0, 3) q. Let us consider [0, m) [0, ), rel intervl nd ny x [0, m) is in [0, ) nd vice vers. For if x = m 1 2 n [0, ) then n [0, m). Thus there exists MOD rel intervl trnsformtion unlike the improperly defined MOD rtionl intervl trnsformtion [0, m) q. We will first illustrte this by some exmples. Exmple 1.8: Let [0,12) be the MOD rel intervl. We define the MOD rel trnsformtion from R = (, ) [0, 12). η r : (, ) [0, 12) s follows: η r ( ) = , η r ( ) = [0, 12), η r ( ) = [0, 12), η r (6.3815) = [0, 12),

20 Renming of Certin Mthemticl Structures 19 η r ( ) = [0, 12) nd η r ( 18.4) = 5.6 [0, 12). This is the wy the rel MOD trnsformtion from the rel line (, ) to [0, 12) is crried out. Exmple 1.9: Let [0, 7) be the rel MOD intervl. η r : (, ) [0, 7) is defined by η r ( ) = [0, 7) η r (10.481) = [0, 7) η r ( 16.5) = 4.5 [0, 7) η r ( ) = [0, 7) η r (35.483) = [0, 7) nd so on. Exmple 1.10: Let [0, 16) be the rel MOD intervl. η r : (, ) [0, 16) η r (48) = 0, η r ( 64) = 0, η r (32.092) = 0.092, η r ( ) = , η r ( ) = nd η r ( ) = re ll in [0, 16).

21 20 Multidimensionl MOD Plnes Thus the MOD rel intervl trnsformtion is well defined. Now we proceed on to define for the complex intervl ( i, i) to [0, m)i F where 0 < m < nd i 2 F = m 1 is defined s the MOD complex finite intervl. We will illustrte this sitution by some exmples. [0, 5)i F, i 2 F = 4 is the complex finite intervl. [0, 17)i F, i 2 F = 16 is gin the complex finite intervl. We will describe the MOD complex trnsformtion. This is illustrted by the following exmple. 2 Exmple 1.11: Let [0, 15)i F, i F = 14 be the MOD complex intervl. Define η : ( i, i) [0, 15)i F s follows: η(8i) = 8i F, η(21i) = 6i F, η(12i) = 12i F, η( 10i) = 5i F nd η( 16i) = 14i F, η r (30.378i) = 0.378i F nd so on. Now we will describe some more exmples. Exmple 1.12: Let [0, 10)i F is the MOD complex intervl. η : ( i, i) [0, 10)i F η(3.585i) = 3.585i F η( i) = i F

22 Renming of Certin Mthemticl Structures 21 η( 3.585i) = 6.415i F η( i) = i F. This is the wy the MOD complex intervl trnsformtion. Clerly [0, m)i F ; 0 < m < 1 is group under ddition; however [0, m)i F is not even closed under product. We will illustrte this by some exmples. Exmple 1.13: Let [0, 12)i F be the MOD complex intervl. η : ( i, i) [0, 12)i F η(25i) = i F, η( 43i) = 5i F, η(50i) = 10i F, η( 50i) = 2i F, η( 25i) = 11i F nd so on. However 0 cts s the dditive identity of [0, 12)i F. Forever x [0, 12)i F we hve unique y [0, 12)i F such tht x + y = 0. Let x = 8.3i F nd y = 4.7i F [0, 12)i F. x + y = i F. Let x = 9.3i F [0, 12)i F there exists y = 2.7i F such tht 9.3i F + 2.7i F = 0.

23 22 Multidimensionl MOD Plnes Exmple 1.14: Let [0, 7)i F be the complex intervl; i 2 F = 6. Let x = 27.8i ( i, i). η(x) = 6.8i F ; Let x = i ( i, i) η(x) = i. Let x = 27.8i ( i, i); η(x) = 0.2i F, η( i) = i F nd so on. This is the wy MOD complex intervl trnsformtion re crried out. Next we study the MOD dul number intervl [0, m)g, g 2 = 0. ( g, g) = {g (, ), g 2 = 0} is the dul number intervl. When [0, m)g = {g [0, m), g 2 = 0} is the MOD dul number intervl. Now we give some exmples. Exmple 1.15: Let [0, 10)g = {g [0, 10), g 2 = 0} be the MOD dul number intervl. Consider η g : ( g, g) [0, 10)g, is defined s follows : η g (108g) = 8g η g ( 108g) = 2g η g (0.258g) = 0.258g

24 Renming of Certin Mthemticl Structures 23 η g ( 0.258g) = 9.742g η g (22.483g) = 2.483g nd so on. Exmple 1.16: Let η g : ( g, g) [0, 9)g be defined by η g (148.3g) = 4.3g, η g ( 148.3g) = 4.7g, η g (5.318g) = 5.318g η g ( 5.318g) = 3.682g η g (0.748g) = 0.748g nd so on. Exmple 1.17: Let [0, 4)g be the MOD dul number intervl. η g : ( g, g) [0, 4)g is defined by η g (428g) = 0, η g (408g) = 0, η g ( 4g) = 0, η g ( 13g) = 3g, η g ( 23g) = g nd so on. Similrly η g (0.8994g) = g nd η g ( g) = g.

25 24 Multidimensionl MOD Plnes Exmple 1.18: Let [0, 9)g be the MOD dul number intervl. η g : ( g, g) [0, 9)g is defined s follows: η g (4.8732g) = g η g ( g) = g η g (12.37g) = 3.37g η g ( 12.37g) = 5.63g nd so on. Infct η g (n9g + tg) = tg where 0 t < 9 η g ( n9g tg) = 9g tg for ll n Z +. η(4.5g) = 4.5g η( 4.5g) = 4.5g nd so on. Exmple 1.19: Let [0, 42)g be the MOD dul number intervl. The MOD dul number intervl trnsformtion; η g : ( g, g) [0, 42)g is defined by η g (43g) = g, η g ( 10g) = 32g, η g (15g) = 15g, η g (2.74g) = 2.74g, η g ( 2.74g) = 39.26g nd so on. Here lso infinitely elements which re periodic re mpped on to sme element of [0, m)g. Infct though [0, m)g itself is infinite still infinite number of elements in ( g, g) re mpped on to single element in [0, m)g. Exmple 1.20: Let G = [0, 8)g be MOD group under ddition. For 0 G serves s the dditive identity.

26 Renming of Certin Mthemticl Structures 25 Infct for every x G there exists unique y G such tht x + y = 0. Let x = g nd y = g G. x + y = g g = g [0, 8)g. Further 0.348g g = 6.468g [0, 8)g. Now {[0, 8)g, +} is MOD belin group of infinite order. Exmple 1.21: Let G = {[0, 5)g, +} be the MOD dul number intervl group of infinite order. Let x = g G, y = g in G is such tht x + y = 0. Thus G, is n belin MOD group of infinite order under ddition modulo 5. This MOD dul number group hs MOD subgroups of both finite nd infinite order. Exmple 1.22: Let S = {[0, 24)g, g 2 = 0, +} be the MOD dul number group under ddition module 24. We see B 1 = {Z 24 g, +} is MOD subgroup of G of finite order. B 2 = {0, 2g, 4g, 6g, 22g} S is lso MOD subgroup of finite order under +. B 3 = {0, 12g} is MOD subgroup of order two. B 4 = {0, 6g, 12g, 18g} is MOD subgroup of order four. B 5 = {0, 8g, 16g} S is gin MOD subgroup of order 3. B 6 = {0, 4g, 8g, 12g, 16g nd 20g} is gin MOD subgroup of order 6.

27 26 Multidimensionl MOD Plnes B 7 = {0, 3g, 6g, 9g, 12g, 15g, 18g, 21g} S is MOD subgroup of order eight. Thus S hs severl MOD subgroups of finite order. Exmple 1.23: Let S = {[0, 7)g, g 2 = 0, +} be MOD group of infinite order. {0, 3.5g} S is MOD subgroup of order two. {0, 1.75g, 3.50g, 5.25g} is MOD subgroup of order four nd so on. So [0, m)g hs MOD subgroups of finite order even though m my be prime or non prime. Alwys [0, m)g hs MOD subgroup of order two, four, five, eight, ten nd so on. Depending on m it will hve MOD subgroups of order 3, 6, 9, 11, 13 nd so on. If in [0, m), m is multiple of 3 then it will hve MOD subgroups of order 3 nd 6. Likewise one cn find conditions for the MOD dul number group [0, m)g to hve finite MOD subgroups of desired order. Next we find [0, m)g the MOD dul number intervl is closed under the product nd the opertion is ssocitive on [0, m)g s g 2 = 0. DEFINITION 1.1: Let S = {[0, m)g g 2 = 0, 0 < m <, } be the MOD dul number semigroup. Since g bg = 0 for ll g, bg [0, m)g. S is defined s the MOD zero squre dul number semigroup of infinite order. Exmples of this sitution is given in the following.

28 Renming of Certin Mthemticl Structures 27 Exmple 1.24: Let S = {[0, 7)g, g 2 = 0, } is the MOD zero squre dul number semigroup which is commuttive nd is of infinite order. For = rg nd b = sg; rg, sg [0, 7)g we hve b = 0 s g 2 = 0. Infct S hs MOD dul number subsemigroups of order two, three, four nd so on. Further ny subset of S is MOD dul number subsemigroup of S. A = {0, 5.02g, g} S is such tht A is MOD dul number subsemigroup of order three. B = {0, g} S is MOD subsemigroup of order two nd so on. Study in this direction is interesting for every subset with zero of S is lso n idel of S. Thus every dul number MOD subsemigroup finite or infinite order is lso MOD dul number idel of S. THEOREM 1.1: Let S = {[0, m)g, g 2 = 0, } be MOD zero squre dul number semigroup. i) Every subset together with zero of S finite or infinite order is dul number MOD subsemigroup of S. ii) Every subset of S with zero finite or infinite order is dul number MOD idel of S. Proof. Directly follows from the fct for x, y S. x y = 0. Hence the clim. Exmple 1.25: Let S = {[0, 12)g, g 2 = 0, } be the MOD dul number semigroup.

29 28 Multidimensionl MOD Plnes Every pir A = {0, x x [0, 12)g \ {0}} is MOD dul number subsemigroup of order two. Every MOD dul subsemigroup of S must contin zero s x y = 0 for every x, y [0, 12)g. Exmple 1.26: Let S = {[0, 15)g, g 2 = 0, } be the MOD dul number semigroup. Every subset A of S with 0 A is MOD dul number subsemigroup s well s n idel n of S. Clerly subset of S which does not contin the zero element is not dul number subsemigroup. Let A = {0.778g, 4.238g, 11.5g, g} S; clerly A is only subset of S nd not dul number subsemigroup of S s 0 A. Now hving defined the notion of MOD dul number semigroup we proceed on to define MOD dul number ring. In the first plce we wish to keep on record tht these MOD dul number semigroups re never monoids s 1 S = {[0, m)g, g 2 = 0, }, 0 < m <. DEFINITION 1.2: Let S = {[0, m)g, +,, g 2 = 0} be MOD dul number ring which hs no unit. Infct S is dul number ring of infinite order which is commuttive. We will first illustrte this sitution by some exmples. Exmple 1.27: Let R = {[0, 9)g, g 2 = 0, +, } be the MOD dul number ring of infinite order. Every subset with zero of R is not MOD dul number subring of R. For if x = 8.312g nd y = 4.095g then xy = 0 = x 2 = y 2.

30 Renming of Certin Mthemticl Structures 29 By x + y = 3.407g {x, y, 0} = A. Hence the clim. Further x + x = 8.312g g = 7.624g A. Similrly y + y = 4.095g g = 8.19g A. Thus we re forced to find under wht condition subset {A, +, } will be subring of MOD dul numbers. If {A, +} is MOD dul number subgroup then {A, +, } is MOD dul number subring. We will give exmples of them. S 1 = {Z 9 g, +, } is zero squre MOD dul number ring. Clerly S 1 is ssocitive. Infct R is itself n ssocitive ring tht is (b c) = ( b) c = 0 for every, b, c R; s R is zero squre ring. S 2 = {0, 4.5g} R is gin zero squre MOD dul number subring. Clerly both S 1 nd S 2 re lso idels of R. Thus every finite MOD dul subring of R is lso MOD idel of R. S 3 = {0, 3g, 6g} R is MOD subring of dul numbers s well s idel of R. Thus these clss of zero squre rings of infinite order behves in very different nd interesting wy. Exmple 1.28: Let R = {[0, 18)g, g 2 = 0, +, } be the MOD dul number ring. Consider S 1 = {0, 9g},

31 30 Multidimensionl MOD Plnes S 2 = {0, 6g, 12g}, S 3 = {0, 3g, 6g, 9g, 12g, 15g}, S 4 = {0, 2g, 4g, 6g, 8g, 10g, 12g, 14g, 16g}, S 5 = {0, 4.50g, 9g, 13.50g}, S 6 = {0, 2.25g, 4.50g, 6.75g, 9g, 11.25g, 13.50g, 15.75g} re ll subgroups of R which re lso MOD subrings of dul numbers. These re lso MOD idels of R. S 7 = {0, 1.8g, 3.6g, 5.4g, 9g, 7.2g, 10.8g, 12.6g, 14.4g, 16.2g} R is MOD dul number subgroup s well s MOD idel of R. S 8 = {0, 1.5g, 3g, 4.5g, 6g, 7.5g, 9g, 10.5g, 12g, 13.5g, 15g, 16.5g} R. S 9 = {0, 1.125g, 2.250g, 3.375g, 4.500g, 5.625g, 6.750g, 7.875g, 9g, g, g, g, 13.5g, g, g, g} R is gin n idel of order 16 however R is of infinite order. Infct R hs infinite number of finite order MOD dul number subrings which re idels. Exmple 1.29: Let R = {[0, 10)g, g 2 = 0, +, } be the MOD dul number ring which is zero squre ring of infinite order. S 1 = {0, 5g}, S 2 = {0, 2g, 4g, 6g, 8g},

32 Renming of Certin Mthemticl Structures 31 S 3 = {0, 2.5g, 5g, 7.50g}, S 4 = {0, 1.25g, 2.5g, 3.75g, 5g, 6.25g, 7.5g, 8.75g}, S 5 = {Z 10 g}, S 6 = {0, 0.5g, 1g, 1.5g, 2g, 2.5g, 3g, 3.5g, 4g, 4.5g, 5g, 5.5g, 6g, 6.5g, 7g, 7.5g, 8g, 8.5g, 9g, 9.5g}, S 7 = {0, 0.625g, 1.25g, 1.875g, 2.5g, 3.125g, 3.750g, 4.375g, 5g, 5.625g, 6.25g, 6.875g, 7.5g, 8.125g, 8.750g, 9.375g} re ll subrings of R which re lso MOD idels of finite order in R. Infct R hs infinitely mny finite MOD dul number subrings which re zero squre subrings of R. Exmple 1.30: Let R = {[0, 7)g, +, } be the MOD dul number zero squre ring. Let S 1 = {0, 3.5g}, S 2 = {0, 1.75g, 3.50g, 5.25g}, S 3 = {0, 1.4g, 2.8g, 4.2g, 5.6g}, S 4 = {0, 0.875g, 1.750g, 2.625g, 3.500g, 4.375g, 5.250, 6.125g} re some of the finite order MOD dul number subgroups which re lso idels of R. Now hving seen both MOD dul number idels of MOD dul number rings nd MOD dul number semigroups we proceed on

33 32 Multidimensionl MOD Plnes to define, describe nd develop the notion of MOD specil dul like number intervl [0, m)h = {h [0, m) nd h 2 = h} is defined s the MOD specil dul like number intervl. Clerly [0, m)h is of infinite order. x 8.7h nd y = 7h [0, m)h then 60.9h (mod m) [0, m)h. We will illustrte this by some exmples. Let [0, 8)h be the MOD specil dul like intervl. Let 4h [0, 8)h; (4h) 2 = 0. Let 0.9h nd 4h [0, 8)h, 3.6h [0, 8)h. 4h + 4h = 0. 4h + 0.9h = 4.9h. This is the wy both the opertions + nd re performed on this MOD specil dul like number intervl. Now we see if ( h, h) is the infinite rel specil dul like number intervl where h 2 = h then we hve the MOD specil dul like number intervl trnsformtions. η h : ( h, h) [0, 8)h s follows: η h (83.5h) = 3.5h, η h (h) = h, η h ( 10.5h) = 5.5h, η h (8h) = 0 = η h (16h) = η h ( 8h) = η h ( 16h) = η h ( 24h) = η h ( 32h) = = η h (n8h); n Z. Thus we see infinite number of points of ( h, h) re mpped on to single point in [0, 8)h.

34 Renming of Certin Mthemticl Structures 33 Let us consider x = 48.5h ( h, h); η h (x) = (7.5h) nd so on. η h (n8h + th) = th for ll t [0, m). η h (n8h th) = mh th for ll th [0, m)h. t is positive vlue. Only t is negtive. Thus we see η h mps infinitely mny periodic elements on to single element. This is the wy η h the MOD specil dul like number intervl trnsformtion functions. Now η h : ( h, h) [0, m)h. h 2 = h, 1 m <. This mp is unique depending only on them. Once the m is fixed η h : ( h, h) [0, m)h is unique. Exmple 1.31: Let η h : ( h, h) [0, 9)h be the MOD specil dul like number trnsformtion. η h (27h) = 0, η h (49h) = 4h, η h (9h) = 0, η h ( 25h) = 2h. Exmple 1.32: Let η h : ( h, h) [0, 12)h be the MOD specil dul like number trnsformtion. η h (27h) = 3h, η h (49h) = h, η h (9h) = 9h,

35 34 Multidimensionl MOD Plnes η h ( 25h) = 2h. We see η h is dependent on [0, m)h cler from the bove two exmples. Now we see the lgebric structure enjoyed by [0, m)h. DEFINITION 1.3: Let [0, m)h = {h [0, m), h 2 = h} be the MOD specil dul like number intervl G = {[0, m)h, +} is defined s MOD specil dul like number intervl group of infinite order. DEFINITION 1.4: S = [0, m)h = { [0, m), h 2 = h} is semigroup under nd it is defined s MOD specil dul like number intervl semigroup. We will give exmples of this sitution. Exmple 1.33: Let M = {[0, 10)h, h 2 = h, +} be MOD specil dul number intervl group of infinite order. M hs MOD specil dul like number intervl subgroups of finite order. S 1 = {Z 10 h, +}, S 2 = {0, 5h, +}, S 3 = {0, 2h, 4h, 6h, 8h, +}, S 4 = {0, 2.5h, 5h, 7.5h}, S 5 = {0, 1.25h, 2.5h, 3.75h, 5h, 6.25h, 7.50h, 8.75h} nd so on re ll finite MOD dul like number finite subgroups of M. Now we see some more exmple.

36 Renming of Certin Mthemticl Structures 35 Exmple 1.34: Let M = {[0, 11)h, h 2 = h, +} be the MOD specil dul like number intervl group of infinite order. S 1 = {Z 11 h, +}, S 2 = {0, 5.5h}, S 3 = {0, 2.75h, 5.50h, 8.75h}, S 4 = {0, 1.375h, 2.750h, 4.125h, 5.500h, 6.875h, 8.250h, 9.625h}, S 5 = {0, 1.1h, 2.2h, 3.3h, 4.4h, 5.5h, 6.6h, 7.7h, 8.8h, 9.9h} re some of MOD specil dul like number intervl finite subgroups of M. Exmple 1.35: Let S = {[0, 12)h, h 2 = h, +} be the MOD specil dul like number group. P 1 = {Z 12 h}, P 2 = {0, 6h}, P 3 = {0, 4h, 8h}, P 4 = {0, 3h, 6h, 9h}, P 5 = {0, 2h, 4h, 6h, 8h, 10h}, P 6 = {0, 2.4h, 4.8h, 7.2h, 9.6h} nd so on re ll MOD specil dul like number subgroups of S of finite order. Next we proceed on to study the notion of MOD specil dul like number semigroups.

37 36 Multidimensionl MOD Plnes Exmple 1.36: Let B = {[0, 9)h, h 2 = h, } be the MOD specil dul like number semigroup. Let x = 0.72h nd y = 0.5h B. x y = 0.72h 0.5h = 0.36h B. x 2 = h y 2 = 0.25h B. This is the wy product is performed on B. Clerly h B cts s the multiplictive identity; for if = xh then xh h = xh = s h 2 = h. Now B is commuttive monoid of infinite order. Next we proceed on to show B = {[0, m)h, } cn hve zero divisors. For if B = {[0, 24)h, h 2 = h, } be the semigroup. Let x = 12h nd y = 2h B, x y = 0. Thus B hs severl zero divisors. x = 3h, y = 8h B is such tht xy = 0 nd so on. Cn B hve idels? the nswer is yes. For tke P 1 = {0, 12h} B is only subsemigroup nd not n idel of B. P 2 = {0, 4h, 12h, 8h, 16h, 20h} B is only subsemigroup nd not n idel.

38 Renming of Certin Mthemticl Structures 37 For if x = 1.1h then x 4h = 4.4h P 2 hence the clim. Let x = 0.9h B nd y = 20h P 2 then x y = 18h P 2. Let P 3 = {0, 2h, 4h,, 22h} be the subsemigroup of B. Let x = 0.5h B nd y = 2h in P 3 then y x = h P 3. Thus P 3 is not n idel of B. Let P 4 = {0, 3h, 6h, 9h, 12h, 15h, 18h, 21h} is only subsemigroup nd not n idel of B. Tke x = 3h P 4, y = 1.5h B, x y = 4.5h P 4. Thus P 4 is not n idel of B. Now hving seen subsemigroups which re not idels it remins s chllenging problem to find idels in B. Will they be only of infinite order or of finite order nd so on? Exmple 1.37: Let B = {[0, 9)h, h 2 = h, } be the MOD specil dul like number monoid of infinite order. B hs severl subsemirings of finite order. However we re not ble to find ny idels of B. In view of this the following is left s n open conjecture. Conjecture 1.1: Let B = {[0, m)h, h 2 = h, } be the MOD specil dul like number commuttive monoid of infinite order. i. Find nontrivil idels of B. ii. Cn idels of B be of finite order?

39 38 Multidimensionl MOD Plnes Exmple 1.38: Let B = {[0, 3)h, h 2 = h, } be the MOD specil dul like number monoid of infinite order. Let P 1 = {0, 1.5h} be subsemigroup nd not n idel. P 2 = {0, 0.75h, 1.50h, 2.25h} is gin MOD subsemigroup nd not n idel of B. Now hving seen exmples of MOD specil dul like number we proceed onto define MOD specil dul like number intervl pseudo rings. DEFINITION 1.5: Let R = {[0, m)h, +,, h 2 = h}; R is defined s the pseudo MOD specil dul like number intervl pseudo ring. Clerly h serves s the multiplictive identity of R. We cll R s pseudo ring s the distributive lw is not true. For, b, c R we hve (b + c) b + c in generl. We will illustrte this sitution by some exmples. Exmple 1.39: Let R = {[0, 5)h, h 2 = h, +, } be the MOD specil dul like number intervl pseudo ring. Consider x = 0.35h, y = 4.15h nd z = 0.85h R. x (y + z) = x (4.15h h) = x 0 Consider x y + x z = 0 I = 0.35h 4.15h h 0.85h = h h II

40 Renming of Certin Mthemticl Structures 39 I nd II re distinct so the distributive lw is not true in R generl. Exmple 1.40: Let R = {[0, 6)h h 2 = h, +, } be the MOD specil dul like number intervl pseudo ring. x = 0.3h, y = 4.8h nd z = 1.2h R is such tht x (y + z) = 0 (s y + z = 0) I x y + x z = 0.3h 4.8h + 0.3h 1.2h = 1.44h h = 1.8h II Clerly I nd II re distinct so the ring is only pseudo nd does not stisfy the distributive lw. Infct R hs nontrivil zero divisors. h is the unit of R. x = 5h is such tht x 2 = h x = 3h nd y = 2h in R is such tht x y = 0, x = 3h nd y = 4h in R is such tht x y = 3h 4h = 0. Now {Z 6 h, +, } is subring which is not n idel of R. Similrly R hs severl subrings which re not idels. Exmple 1.41: Let M = {[0, 7)h, h 2 = h, +, } be the MOD specil dul like number intervl pseudo ring. M hs zero divisors. M hs subring however we re not in position to find idels in these pseudo ring.

41 40 Multidimensionl MOD Plnes We leve it s open problem to find MOD idels of specil dul like number intervls. Conjecture 1.2: Let R = {[0, m)h, h 2 = h, +, } be the MOD specil dul like number intervl pseudo ring. i. Cn R hve idels of finite order? ii. Find nontrivil idels of R. Now hving seen pseudo ring of specil dul like number intervls we next proceed on to discuss bout the specil qusi dul number intervls ( k, k) where k 2 = k. Now [0, m)k = {k [0, m), k 2 = (m 1)k} is defined s the MOD specil qusi dul number intervl. We define MOD specil qusi dul number intervl nd the MOD specil qusi dul number intervl trnsformtion. Let η k : ( k, k) [0, m)k where k 2 = k in ( k, k) nd k 2 = (m 1)k; in cse of intervl [0, m)k. η k (k) = k. We will illustrte this by some exmples. Exmple 1.42: Let ( k, k) be the rel specil qusi dul numbers. [0, 5)k be the specil qusi dul number intervl. Define η k : ( k, k) [0, 5)k η k (4.8222k) = k η k ( k) = k η k ( k) = k η k ( k) = k η k (108.3k) = 3.3k

42 Renming of Certin Mthemticl Structures 41 η k ( 108.3k) = 1.7k. η k (5k) = η k (10k) = η k (15k) = η k (45k) = 0 thus severl; infct infinitely mny elements re mpped onto single element. η k (5k + 0.3k) = η k (10k + 0.3k) = η k (15.3k) = η k (20.3k) = η k (25.3k) = η k (30.3k) = η k (35.3k) = η k (40.3k) = η k (45.3k) = = η k (5nk + 0.3k) = 0.3k. Thus infinitely mny numbers ( k, k) is mpped on to single element. η k (6.35k) = 1.35k, η k ( 6.35k) = 3.65k nd so on. Exmple 1.43: Let η k : ( k, k) [0, 11)k, k 2 = k in rel nd k 2 = 10k in [0, 11)k. η k ( k, k) [0, 11)k. η k (10k) = 10k, η k ( 48k) = 3k, η k (48k) = 8k, η k ( 110k) = 0, η k (49k) = 5k nd so on. Now s in cse of ll other MOD intervl trnsformtions this η k is the sme s tht of other MOD trnsformtion. Now we proceed on to define lgebric structures on [0, m)k, k 2 = (m 1)k.

43 42 Multidimensionl MOD Plnes DEFINITION 1.6: Let B = {[0, m)k k 2 = (m 1)k, +} be the group under ddition modulo m. B is n infinite commuttive MOD specil qusi dul number intervl group. We will give exmples of them. Exmple 1.44: Let G = {[0, 8)k, k 2 = 7k, +} be the MOD specil qusi dul like number intervl group. G hs severl subgroups of finite order. B 1 = {0, 2k, 4k, 6k}, B 2 = {0, 4k}, B 3 = {0, 0.8k, 1.6k, 2.4k, 3.2k, 4k, 4.8k, 5.6k, 6.4k, 7.2k}, B 4 = {0, 1.6k, 3.2k, 4.8k, 6.4k}, B 5 = {k, 2k,, 7k, 0}, B 6 = {0, 0.5k, 1k, 1.5k, 2k, 2.5k, 3k, 3.5k, 4k, 4.5k, 5k, 5.5k, 6k, 6.5k, 7k, 7.5k}, B 7 = {0, 0.25k, 0.5k, 0.75k, k, 1.25k, 1.50k, 1.75k, 2k, 2.25k, 2.5k, 2.75k, 3k, 3.25k, 3.5k, 3.75k,, 7k, 7.25k, 7.5k, 7.75k} nd so on re ll MOD subgroups of finite order. We cn hve severl such MOD subgroups of finite order. Exmple 1.45: Let G = {[0, 15)k, k 2 = 14k, +} be the MOD specil qusi dul number intervl group. Let P 1 = Z 15 k, P 2 = {0, 3k, 6k, 9k, 12k},

44 Renming of Certin Mthemticl Structures 43 P 3 = {0, 5k, 10k}, P 4 = {0, 7.5k}, P 5 = {0, 3.75k, 7.50k, 11.25k}, P 6 = {0, 2.5k, 5k, 7.5k, 10k, 12.5k}, P 7 = {0, 1.875k, 3.750k, 5.625k, 7.5k, 9.375k, k, k}, P 8 = {0, 1.5k, 3k, 4.5k, 6k, 7.5k, 9k, 10.5k, 12k, 13.5k}, P 9 = {0, 1.25k, 2.5k, 3.75k, 5k, 6.25k, 7.5k, 8.75k, 10k, 11.25k, 12.5k, 13.75k} nd P 10 = {0, k, k, k, k, k, k, k, 7.500k, k, k, k, k, , , k} re ll some of the MOD subgroups of infinite order. Infct G hs infinite number of finite MOD subgroups. Exmple 1.46: Let G = {[0, 24)k, k 2 = 23k, +} be the MOD specil qusi dul number intervl group of infinite order. G hs severl finite order MOD subgroups. Infct H 1 = 0.1k is MOD subgroup of G generted by 0.1k. Likewise H 2 = 0.01k is gin MOD subgroup of G. H 3 = { 0.001k } is gin MOD subgroup of G. Likewise we cn hve severl MOD subgroups of finite order in G.

45 44 Multidimensionl MOD Plnes Next we proceed on to describe nd define the notion of MOD specil qusi dul number intervl semigroup of infinite order under product. DEFINITION 1.7: Let S = {[0, m)k, k 2 = (m 1)k, } be the MOD specil qusi dul number intervl semigroup of infinite order. However S is commuttive MOD semigroup or to be more precise S is MOD specil qusi dul number intervl semigroup of infinite order. We will illustrte this sitution by some exmples. Exmple 1.47: Let S = {[0, 5)k, k 2 = 4k, } be the MOD specil qusi dul number intervl semigroup. P 1 = {Z 5 k, } is subsemigroup of finite order. Cn S hve ny other subsemigroup of finite order? Let P 2 = 0.1k ; clerly P 2 is n infinite order MOD subsemigroup of S. S. P 3 = 0.01k is gin n infinite order MOD subsemigroup of Study in this direction is innovtive nd interesting. The following is left s n open problem. Problem 1.1: Let S = {[0, m)k, k 2 = (m 1)k, } be the MOD specil qusi dul number intervl semigroup. If m is prime cn S hs more thn one finite order subsemigroup got from S. Exmple 1.48: Let S = {[0, 11)k, k 2 = 10k, } be the MOD specil qusi dul number intervl semigroup. P 1 = Z 11 k is MOD subsemigroup of order 11.

46 Renming of Certin Mthemticl Structures 45 P 2 = {10k} is MOD subsemigroup of S. P 3 = {10k, k} is lso MOD subsemigroup of S. P 4 = {0, 10k} is MOD subsemigroup of order two. Interested reder cn study both finite nd infinite order MOD subsemigroups of S. Clerly the MOD specil qusi dul number intervl semigroup hs zero divisors. THEOREM 1.2: Let S = {[0, m)k, k 2 = (m 1)k, } be the MOD specil qusi dul number semigroup. If m is composite number nd Z m k hs lrge number of subsemigroups then S hs mny MOD specil qusi dul number subsemigroups of finite order. Proof of the theorem is direct nd hence left s n exercise to the reder. Finding idempotents of S is chllenging problem. Next we proceed on to define the notion of MOD specil qusi dul number intervl pseudo rings. DEFINITION 1.8: R = {[0, m)k, k 2 = (m 1)k, +, } is defined s the MOD specil qusi dul number intervl pseudo ring s the opertion of + over is not distributive. Tht is (b + c) b + c in generl for, b, c R. We will illustrte this by some exmples. Exmple 1.49: Let R = {[0, 9)k, k 2 = 8k, +, } be the MOD specil qusi dul number pseudo ring. Let x = 8.3k, y = 6.107k nd z = 0.5k R;

47 46 Multidimensionl MOD Plnes x (y + z) = 8.3k(6.107k + 0.5k) = 8.3k 6.607k = k 8 = k I Consider x y + x z = 8.3k 6.107k + 8.3k 0.5k = k k = k + 6.2k For this triple the distributive lw is true. = k II Consider x = 8.012k, y = 4.03k nd z = 4.97k R Consider x (y + z) = 8.012k (4.03k k) = 8.012k 0 = 0 I x y + x z = 8.012k 4.03k k 4.97k = k k = k k = 0.864k II

48 Renming of Certin Mthemticl Structures 47 I nd II re distinct so for this triple the distributive lw is not true. Infct there re infinitely mny such triples such tht the distributive lw is not true. Now we hve seen the pseudo rings of MOD specil qusi dul number re non distributive but is of infinite order. Certinly this MOD pseudo rings hve zero divisors, units nd idempotents solely depending on the m of the MOD intervl [0, m)k. Thus we hve six types of MOD intervls, rels [0, m), complex modulo integer [0, m)i F, neutrosophic MOD intervl [0, m)i, dul number MOD intervl [0, m)g, g 2 = 0, specil dul like number MOD intervl [0, m)h, h 2 = h specil qusi dul number MOD intervl [0, m)k, k 2 = (m 1)k. On ll these structures MOD groups, MOD semigroups nd MOD pseudo rings is defined nd developed. Only in cse of MOD specil dul number intervl [0, m)g, g 2 = 0 we see both the semigroups nd rings re zero squre semigroups nd rings respectively. Only in this cse the distributive lw is lso true. We will illustrte mtrices nd supermtrices built using these structures by exmples. Exmple 1.50: Let A = {(x 1, x 2, x 3, x 4 ) x i [0, 6); +, 1 i 4} be the collection of ll row mtrices. We cn define on A two types of opertions + nd. Under + A is group defined s the MOD rel row mtrix group. A is commuttive nd is of infinite order.

49 48 Multidimensionl MOD Plnes Exmple 1.51: Let B = {(x 1, x 2, x 3, x 4, x 5, x 6, x 7 ) x i [0, 17); +, 1 i 7} be the collection of ll 1 7 row mtrices. B is MOD rel row mtrix group. Exmple 1.52: Let C = i [0, 15); 1 i 6, +} be MOD rel column mtrix intervl group. Exmple 1.53: Let M = i [0, 23), 1 i 10, +} be the MOD rel intervl column mtrix group of infinite order.

50 Renming of Certin Mthemticl Structures 49 Exmple 1.54: Let V = i [0, 43), 1 i 7, +} be the MOD rel intervl column mtrix group of infinite order. Exmple 1.55: Let V = i [0, 45); 1 i 12, +} be the MOD rel intervl squre mtrix group of infinite order. Exmple 1.56: Let M = i [0, 15); 1 i 12, +} be the MOD rel mtrix intervl group of infinite order.

51 50 Multidimensionl MOD Plnes Exmple 1.57: Let W = i [0, 14); 1 i 12; +} be the MOD rel number intervl group of infinite order. This hs subgroups of both finite order nd infinite order. Exmple 1.58: Let S = i [0, 47); 1 i 18, +} be the MOD rel intervl mtrix group. This hs t lest 18 C C C 18 number of finite subgroups. Study of using other MOD intervls is considered s mtter of routine nd left s n exercise to the reder however one or two exmples to illustrte this is given. Exmple 1.59: Let W = {(x 1, x 2,,, x 7 ) x i [0, 9)i F ; 1 i 7, i 2 F = 8, +} be the MOD complex modulo integer row mtrix group.

52 Renming of Certin Mthemticl Structures 51 W is of infinite order hs subgroups of finite nd infinite order. Exmple 1.60: Let V = x1 x2 x3 x x x x i [0, 11)i F ; i 2 F = 10, 1 i 6, +} be the MOD complex modulo integer mtrix group. V hs subgroups of both finite nd infinite order. Exmple 1.61: Let X = i [0, 43)i F ; i 2 F = 42, 1 i 15, +} be the MOD complex modulo integer intervl group. X is of infinite order. Exmple 1.62: Let Y = i [0, 15)i F ; i 2 = 14, F 1 i 24, +} be the MOD complex modulo integer intervl group. Y is of infinite order.

53 52 Multidimensionl MOD Plnes Exmple 1.63: Let M = i [0, 25)i F ; i 2 F = 24, 1 i 20; +} be the MOD complex modulo integer intervl group of infinite order. Exmple 1.64: Let W = i [0, 43)i F, i 2 = 42, F 1 i 16, +} be the MOD complex modulo integer intervl group of infinite order. Exmple 1.65: Let M = {( 1, 2,, 9 ) i [0, 18)I; I 2 = I, 1 i 9, +} be the MOD neutrosophic modulo integer group. Exmple 1.66: Let X = i [0, 98)I; I 2 = I, 1 i 18, +} be the MOD neutrosophic modulo integer intervl group.

54 Renming of Certin Mthemticl Structures 53 Exmple 1.67: Let M = i [0, 14)I; I 2 = I, 1 i 12, +} be the MOD modulo integer neutrosophic mtrix group. Exmple 1.68: Let A = i [0, 31)I; I 2 = I, 1 i 9; +} be the MOD modulo integer neutrosophic mtrix group. Exmple 1.69: Let T = i [0, 14)I; I 2 = I, 1 i 15, +} be the MOD modulo integer neutrosophic intervl mtrix group.

55 54 Multidimensionl MOD Plnes Exmple 1.70: Let S = i [0, 10)I; I 2 = I, 1 i 12, +} be the MOD modulo integer neutrosophic intervl group. Exmple 1.71: Let M = i [0, 17)g, g 2 = 0, 1 i 6, +} be the MOD modulo integer dul number intervl mtrix group. Exmple 1.72: Let V = i [0, 18)g, g 2 = 0, 1 i 6, +} be the MOD dul number intervl mtrix group.

56 Renming of Certin Mthemticl Structures 55 Exmple 1.73: Let M = i [0, 17)g, g 2 = 0; 1 i 18, +} be the MOD dul number intervl mtrix group. Exmple 1.74: Let V = i [0, 29)g, g 2 = 0; 1 i 27, +} be the MOD dul number group under +. Exmple 1.75: Let M = {[ 1 10 ] i [0, 19)h; h 2 = h, 1 i 10; +} be the MOD specil dul like number intervl mtrix group under +. Exmple 1.76: Let T = {( 1 19 ) i [0, 43)h, h 2 = h; 1 i 19, +} be the MOD specil dul like number intervl mtrix group under +.

57 56 Multidimensionl MOD Plnes Exmple 1.77: Let M = i [0, 14)h; h 2 = h; 1 i 16, +} be the MOD specil dul like number intervl mtrix group. Exmple 1.78: Let T = i [0, 178)h, h 2 = h; 1 i 15, +} be the MOD specil dul like number intervl mtrix group. Exmple 1.79: Let S = {( 1 9 ) i [0, 43)k; k 2 = 42k; 1 i 9; +} be the MOD specil qusi dul number intervl mtrix group. Exmple 1.80: Let M = i [0, 18)k, k 2 = 17k, 1 i 9, +} be the MOD specil qusi dul number intervl mtrix group.

58 Renming of Certin Mthemticl Structures 57 M is infinite nd hs subgroups of finite order. Exmple 1.81: Let T = i [0, 11)k, k 2 = 10k, 1 i 24, +} be the MOD specil qusi dul number intervl mtrix group. M hs t lest n( 24 C C C 24 ) number of subgroups where n is the number of finite subgroups of T. Next we proceed on to give exmples of MOD mtrix intervl semigroups using these 6 types of MOD intervls. Exmple 1.82: Let B = {(x 1, x 2, x 3 ) x i [0, 24); 1 i 3; } be the MOD rel intervl mtrix semigroup. B is commuttive nd is of infinite order. B hs infinite number of zero divisors. B hs lso idempotents nd (1, 1, 1) cts s the multiplictive identity.

59 58 Multidimensionl MOD Plnes Exmple 1.83: Let M = i [0, 23); 1 i 5, n } be the MOD rel intervl mtrix semigroup. M hs infinite number of zero divisors, units nd idempotents. Exmple 1.84: Let P = i [0, 108); 1 i 9, n } be the MOD rel intervl mtrix commuttive semigroup. P hs infinitely mny zero divisors nd severl subsemigroups. If in the bove exmple n is replced by the usul mtrix product then P is not commuttive semigroup. P is only MOD rel intervl mtrix non commuttive monoid of infinite order.

60 Renming of Certin Mthemticl Structures 59 Exmple 1.85: Let W = i [0, 17); 1 i 14, n } be the MOD rel mtrix intervl semigroup. W hs infinite number of zero divisors. However W hs only finite number of idempotents nd units. Next we proceed on to give exmples to show MOD intervl mtrix finite complex modulo integer semigroups does not exist. Exmple 1.86: Let W = {( 1, 2, 3, 4, 5 ) i [0, 7)i F ; i 2 F = 6; 1 i 5, } be the MOD intervl mtrix finite complex modulo integer, W is not even closed under.

61 60 Multidimensionl MOD Plnes Exmple 1.87: Let M = i [0, 8)i F ; i 2 = 7, 1 i 6, F n} cnnot be the MOD complex modulo integer intervl semigroup s it is not defined. Exmple 1.88: Let V = i [0, 19)i F, i 2 = 18; 1 i 9, F n} be the MOD complex modulo integer intervl which is not closed. Exmple 1.89: Let M = {( 1, 2,, 9 ) where i [0, 15)I; 1 i 9, ; I 2 = I} be the MOD neutrosophic intervl semigroup of infinite order. (I I I I I) cts s the identity. So M is commuttive monoid of infinite order.

62 Renming of Certin Mthemticl Structures 61 Exmple 1.90: Let W = i [0, 42)I; 1 i 13, I 2 = I, n } be the MOD neutrosophic intervl monoid of infinite order. I I I = Id is the identity element of W; W =. Exmple 1.91: Let V = i [0, 45)I; I 2 = I, 1 i 4, n } be the MOD neutrosophic intervl mtrix semigroup. V is the commuttive monoid. Id = I I I I is the identity element of V. If in V the nturl product n is replced by then V is I 0 non commuttive monoid nd 0 I = I 2 2 is the identity of V. In both cses V hs zero divisors, units nd idempotents.

63 62 Multidimensionl MOD Plnes Exmple 1.92: Let B = i [0, 15)I; I 2 = I, 1 i 12, n } be the MOD neutrosophic intervl commuttive semigroup (infct monoid) of infinite order. I I I I I I Id = I I I I I I is the identity of B. Exmple 1.93: Let T = i [0, 13)I; I 2 = I,1 i 24, n } be the MOD neutrosophic intervl semigroup (monoid) of T. T is of infinite order nd is commuttive.

64 Renming of Certin Mthemticl Structures 63 Id = I I I I I I I I I I I I I I I I I I I I I I I I is the identity of T. T hs infinite number of zero divisors, finite number of units nd idempotents. T hs mny subsemigroups of finite order s well s of infinite order. Exmple 1.94: Let B = i [0, 20)g, g 2 = 0; 1 i 9, n } be the MOD dul number intervl semigroup. B is not monoid. Infct B is zero squre semigroup. B hs both finite nd infinite subsemigroups. Exmple 1.95: Let W = {( 1, 2, 3, 4, 5, 6, 7, 8 ) i [0, 11)g, g 2 = 0; 1 i 8, } be the MOD dul number intervl semigroup of infinite order. W is commuttive nd W is zero squre semigroup. Infct every subsemigroup is lso n idel of W.

65 64 Multidimensionl MOD Plnes Exmple 1.96: Let S = i [0, 15)g, g 2 = 0; 1 i 9, n } be the MOD dul number intervl semigroup of infinite order. S is zero squre semigroup nd hs no unit. S is commuttive. If n, the nturl product is replced by then lso S is commuttive s S is zero squre semigroup. Exmple 1.97: Let P = {( 1, 2, 3, 4, 5, 6, 7 ) i [0, 9)h; h 2 = h; 1 i 7, } be the MOD specil dul like number semigroup of infinite order. Id = (h, h, h, h, h, h) P cts s the identity. So P is infct commuttive monoid of infinite order. Exmple 1.98: Let M = i [0, 15)h, h 2 = h, n ; 1 i 10} be the MOD specil dul like number intervl semigroup.

66 Renming of Certin Mthemticl Structures 65 Id = h h h is the identity element of M for nd X M, X n Id = Id n X = X. Infct M is n infinite intervl commuttive monoid. We will give few more exmples of this structure. Exmple 1.99: Let M = i [0, 18)h; h 2 = h; 1 i 10, n } be the MOD specil dul like number intervl semigroup of infinite order. Infct M is only monoid Id = element of M. h h h h h h h h h h M is the identity Exmple 1.100: Let P = {( 1, 2, 3, 4, 5, 6, 7, 8, 9 ) i [0, 19)k, k 2 =18k, n ; 1 i 9}

67 66 Multidimensionl MOD Plnes be the MOD specil qusi dul number intervl semigroup of infinite order. Id = (18k, 18k,, 18k) cts s the identity of P. So P is the monoid of infinite order. Exmple 1.101: Let M = i [0, 48)k, k 2 = 47k, 1 i 6, n } be the MOD specil qusi dul number intervl semigroup of infinite order. Id = 47k 47k 47k 47k 47k M cts s the identity of M. 8k 10k Let A = 46k M. 0 11k

68 Renming of Certin Mthemticl Structures 67 A n Id = 8k 10k 46k 0 11k n 47k 47k 47k 47k 47k 8k 10k = 46k 0 11k = A. Exmple 1.102: Let M = i [0, 12)k, k 2 = 11k, 1 i 24, n } be the MOD specil qusi dul number intervl semigroup of infinite order. M is infct commuttive monoid. Id = 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k 11k A n Id = Id n A = A for ll A M. is the identity of M. Exmple 1.103: Let D = i [0, 24)k, k 2 = 23k, 1 i 9, n }

69 68 Multidimensionl MOD Plnes be the MOD commuttive specil qusi dul number intervl semigroup (monoid). 23k 23k 23k Id = 23k 23k 23k 23k 23k 23k is the identity of D. If in D we replce by then D is non commuttive monoid 23k 0 0 with I 3 3 = 0 23k k Thus for ll A D, A I 3 3 = I 3 3 A = A. Exmple 1.104: Let B = i [0, 6)k, k 2 = 5k, 1 i 12; n } be MOD commuttive specil qusi dul number intervl semigroup. 5k 5k 5k Infct Id = 5k 5k 5k 5k 5k 5k 5k 5k 5k is the identity element of B.

70 Renming of Certin Mthemticl Structures 69 So B is commuttive monoid of infinite order. Next we just give exmples of MOD intervl pseudo rings using ll the 6 MOD intervls. Exmple 1.105: Let M = {(x 1, x 2, x 3, x 4, x 5, x 6, x 7 ) x i [0, 8); 1 i 7, +, } be MOD rel intervl pseudo ring. M is commuttive nd is of infinite order. Id = (1, 1, 1, 1, 1, 1, 1) in M cts s the identity element of M. M hs severl pseudo subring of infinite order which re idels. M. Infct M hs subrings of finite order which re not idels of Exmple 1.106: Let T = i [0, 17); 1 i 5, +, n } be the MOD rel intervl commuttive pseudo ring with identity; Id = T. 1 1

71 70 Multidimensionl MOD Plnes This T hs pseudo idels of infinite order nd ll subrings of finite order re not idels of T. Exmple 1.107: Let M = i [0, 144); 1 i 4, +, n } be the MOD rel intervl pseudo ring which is commuttive nd hs Id = is the identity M is of infinite order. Now if in M n is replced by ; then M is MOD rel intervl pseudo ring which is non commuttive. I 2 2 = product M is the identity of M under the usul Exmple 1.108: Let T = i [0, 45); 1 i 24, +, n }

72 Renming of Certin Mthemticl Structures 71 be the MOD rel intervl pseudo ring. T hs infinite number of zero divisors. This ring hs only infinite order subrings to be idels. However ll infinite order subrings of T re not idels of T. For P = , 2, 3 Z 45, i [0, 45), 4 i 12} is n infinite order pseudo subring which is not n idel of T. However none of the finite subrings re idel of T. L = , 2, 3 [0, 45), +, n } is MOD intervl pseudo subring of T.

73 72 Multidimensionl MOD Plnes L is n idel nd L is infinite. Exmple 1.109: Let W = i [0, 16); 1 i 16, +, n } be the MOD rel intervl pseudo ring. W is commuttive nd is of infinite order. Id = W cts s the identity of W. W is pseudo ring with identity. W hs idel of infinite order. None of the pseudo subrings of finite order of W is n idel of W. Now if n the nturl product is replced by the usul product then W is non commuttive MOD rel intervl pseudo ring. I 4 4 = cts s the identity of W

74 Renming of Certin Mthemticl Structures 73 Exmple 1.110: Let M = {( 1, 2, 3, 4, 5, 6 ) i [0, 16); 1 i 6, +, } be the MOD rel modulo integer intervl pseudo ring. M is commuttive pseudo ring with Id = (1, 1, 1, 1, 1, 1). M hs zero divisors. M hs idels ll of which re of infinite order. M hs subrings of finite order which re not idels of M. All idels of M re of infinite order. Exmple 1.111: Let V = i [0, 43); 1 i 6; +, n } be the MOD rel integer intervl pseudo ring which is commuttive nd hs no identity. V hs pseudo idels of infinite order. P = [0, 43), +, n } V is n idel of V.

75 74 Multidimensionl MOD Plnes V is of infinite order. V hs 6 C C C C C 5 re ll idels of V. Similrly V hs 6 C C C C C C 6 number of subrings. Exmple 1.112: Let M = i [0, 13); +, n } be the MOD rel intervl pseudo ring. M hs infinite number of zero divisors. Only M hs finite MOD subrings which re not idels. M hs idels of infinite order given by P 1 = P 2 = [0, 13); +, n }, 2 [0, 13), +, n } nd so on ll re pseudo idels of M. P 3 = [0, 13), +, n },,

76 Renming of Certin Mthemticl Structures 75 nd so on. P 9 = P 1, 2,, 8 = [0, 13), +, n },, i [0, 13); 1 i 8, +, n } P 2,, 9 = i [0, 13); 2 i 9, +, n } re ll pseudo idels of M. Thus M hs 9 C C C C 8 number of pseudo idels ll of them re of infinite order. Exmple 1.113: Let L = {( 1, 2,, 6 ) i [0, 15)I; I 2 = I, 1 i 6, +, } be the MOD neutrosophic intervl pseudo ring of infinite order. L hs infinite number of zero divisors. L hs finite subrings nd they re not idels. L hs infinite order pseudo subrings which re idels. M 1 = {( 1, 0,, 0) 1 [0, 15)I, +, }, M 2 = {(0, 2, 0,, 0) 2 [0, 15)I, +, }, M 3 = {(0, 0, 3, 0, 0, 0) 3 [0, 15)I, +, },,

77 76 Multidimensionl MOD Plnes M 1, 2,, 5 = {( 1, 2, 3, 4, 5, 0) i [0, 15)I; 1 i 5},, M 2, 3,, 6 = {(0, 2, 3, 4, 5, 6 ) i [0, 15)I; 2 i 6, +, } re ll pseudo subrings which re ll pseudo idels of L. Exmple 1.114: Let M = i [0, 160)I; I 2 = I, 1 i 5; +, n } be the MOD neutrosophic intervl pseudo ring. M hs infinite number of zero divisors. M hs 5 C C C C 4 number of MOD neutrosophic intervl pseudo idels. Exmple 1.115: Let M = i [0, 14)I, I 2 = I, 1 i 8, +, n } be the MOD neutrosophic intervl pseudo ring. M hs infinite number of zero divisors. M hs finite number of finite subrings. M hs pseudo idels of infinite order. All pseudo idels cn only be of infinite order. Exmple 1.116: Let

78 Renming of Certin Mthemticl Structures 77 P = i [0, 15)I; I 2 = I, 1 i 9; +, n } be the MOD neutrosophic intervl pseudo ring. P hs infinite number of zero divisors. P hs subrings of finite order which re not idels. All pseudo idels of P re of infinite order. Exmple 1.117: Let M = i [0, 16)I; I 2 = I, 1 i 15, +, n } be the MOD neutrosophic intervl pseudo ring. M hs infinite number of zero divisors. M hs finite number of idempotents nd so on. Exmple 1.118: Let B = {( 1, 2, 3, 4, 5, 6 ) i [0, 10)g, g 2 = 0; 1 i 6; +, } be the MOD dul number intervl ring. Clerly these rings re zero squre rings so the distributive lw is true. B =. B hs idels of finite order. B hs infinite number of idels.

79 78 Multidimensionl MOD Plnes B lso hs infinite number of subrings ll of which re only idels. Exmple 1.119: Let M = i [0, 40)g, g 2 = 0; 1 i 7, +, n } be the MOD dul number intervl ring which is not pseudo. M hs infinite number of idels both of finite nd infinite order. M is zero squre ring. Exmple 1.120: Let M = i [0, 12)g, g 2 = 0; 1 i 30, +, n } be the MOD dul number ring which is not pseudo. Every subset of M with zero mtrix is gin n idel s well s subring. In view of ll these we hve the following theorem. THEOREM 1.3: Let S = {A = (m ij ) m ij [0, m)g; g 2 = 0; 1 i n nd 1 j s, +, n } be n s mtrix MOD dul number ring.

80 Renming of Certin Mthemticl Structures 79 i. S is zero squre ring. ii. All subsets P which re groups under + re subrings s well s idels. iii. S hs infinite number of idels. iv. All subrings of S re idels of S. Proof. Follows from the fct S is zero squre ring. Now we proceed on to describe the MOD specil dul like number pseudo ring by exmples. Exmple 1.121: Let M = {( 1,, 7 ) i [0, 12)h, h 2 = h; 1 i 7, +, } be the MOD specil dul like number intervl pseudo ring. M is of infinite order. M hs infinite number of zero divisors. M hs subrings of finite order which re not idels. M hs pseudo idels which re of infinite order. Let P 1 = {( 1, 0,, 0) 1 [0, 12)h, h 2 = h, +, }, P 2 = {(0, 2, 0,, 0) 2 [0, 12)h; h 2 = h, +, }, on. P 3 = {(0, 0, 3, 0,, 0) 3 [0, 12)h, h 2 = h, +, } nd so P 7 = {(0, 0,, 0, 7 ) 7 [0, 12)h, h 2 = h,, +}, P 1,2 = {( 1, 2, 0,, 0) 1, 2 [0, 12)h; h 2 = h, +, },, P 1,7 = {( 1, 0, 0, 0, 0, 0, 7 ) 1, 7 [0, 12)h; h 2 = h, +, }, P 1, 2, 3 = {( 1, 2, 3, 0,, 0) 1, 2, 3 [0, 12)h; h 2 = h, +, },,

81 80 Multidimensionl MOD Plnes P 1, 6, 7 = {( 1, 0,, 0, 6, 7 ) 1, 6, 7 [0, 12)h; h 2 = h, +, } nd so on. P 5, 6, 7 = {(0, 0, 0, 0, 5, 6, 7 ) 5, 6, 7 [0, 12)h, h 2 = h, +, }, P 1, 2, 3, 4 = {( 1, 2, 3, 4, 0, 0, 0) i [0, 12)h, h 2 = h, +, ; 1 i 4} nd so on re ll pseudo subsemirings which re lso pseudo idels of M. M hs t lest 5( 7 C C C 7 ) number of finite subrings which re not idels of M. M hs t lest 7 C C C 6 number of pseudo subrings of infinite order which re idels. Infct M hs more thn 7 C C C 6 number of pseudo infinite subsemirings which re not idels of M. Exmple 1.122: Let B = i [0, 17)h, h 2 = h, 1 i 6, +, n } be the MOD specil dul like number intervl pseudo rings. B hs infinite number of zero divisors. B lso hs finite number of finite order subrings which re not idels.

82 Renming of Certin Mthemticl Structures M 1 = { 0 1 Z 17 h, h 2 = h, +, n } is only subring of finite order nd not n idel. 1 0 M 2 = { 0 1 [0, 17)h, h 2 = h, +, n } is pseudo subsemiring s well s pseudo idel of B. 1 2 Let T 1 = { Z 17 h nd 2 [0, 17)h, h 2 = h, +, n } be the MOD pseudo subsemiring of B of infinite order. Clerly T 1 is not n idel. So B hs mny pseudo subsemirings of infinite order which re not idels of B. Exmple 1.123: Let D = i [0, 12)h, h 2 = h, 1 i 9, +, n } be the MOD specil dul like number pseudo intervl ring of infinite order.

83 82 Multidimensionl MOD Plnes D hs pseudo idels ll of which re of infinite order. D hs subrings which re not idels of finite order. D hs infinite pseudo subrings which re not idels M 1 = { [0, 12)h, h 2 = h, +, n } is pseudo subring which is lso n idel of D M 2 = { Z 12 h; h 2 = h, +, n } is subring which is not pseudo idel nd M 2 is of finite order M 3 = { Z 12 h, 2 [0, 12)h; h 2 = h, +, n } is pseudo subring. But M 3 is not pseudo idel only pseudo subring. Infct D hs mny infinite order pseudo subrings which re not pseudo idels. However ll mtrix MOD specil dul like number intervl pseudo rings hs MOD pseudo idels ll of them re of infinite order but hs finite subrings s well s infinite order pseudo subrings which re not pseudo idels. Next we wish to stte MOD specil dul like number intervl pseudo ring is Smrndche ring if Z m I (Z m i F or Z n k or Z n h) is Smrndche ring.

84 Renming of Certin Mthemticl Structures 83 This is true for ll MOD pseudo rings except the MOD specil dul number intervl ring. R = {[0, m)g, g 2 = 0, +, } for they re in the first plce not pseudo ring for they stisfy the distributive lw nd secondly they re such tht b = 0 for ny, b [0, m)g tht is R is zero squre ring. Finlly we just give few illustrtions of MOD specil qusi dul number intervl pseudo rings. B = {[0, m)k k 2 = (m 1)k, +, }. Exmple 1.124: Let B = {(x 1, x 2, x 3 ) x i [0, 9)k, k 2 = 8k, 1 i 3, +, } be the MOD specil qusi dul number pseudo ring. B hs zero divisors. Clerly (8k, 8k, 8k) cts s the identity of mtrix ring with entries from Z 9 k. For if x = (3k, 5k, 7k) B x (8k, 8k, 8k) = (24 k 2, 40k 2, 56k 2 ) = (6 8k, 4 8k, 2 8k) = (3k, 5k, 7k) = x B. Thus we see for B contins both subrings of finite order s well s infinite pseudo subrings which re pseudo idels. However P = {(x 1, x 2, 0) x 1 Z 9 k nd x 2 [0, 9)k, +, } is only pseudo subring of infinite order but is not n idel.

85 84 Multidimensionl MOD Plnes For if A = (0.2k, 0.9k, 6k) P then for x = (7k, 0.8k, 0) P. Ax = (0.2k, 0.9k, 6k) (7k, 0.8k, 0) = (1.4 8k, 0.72k 8, 0) = (2.2k, 5.76k, 0) P. Thus P is only pseudo subring nd not n idel of B. Hence there re pseudo infinite subrings which re not pseudo idels. Exmple 1.125: Let 1 M = { 2 6 i [0, 14)k, k 2 = 13k, +, n } be the MOD specil qusi dul number intervl pseudo ring. M hs zero divisors. Finding idempotents is difficult tsk. M hs t lest 2( 6 C C 6 ) number of finite subrings nd hs t lest 6 C C C 5 number of infinite subrings which re idels. M hs mny infinite order subrings which re not idels.

86 Renming of Certin Mthemticl Structures 85 For instnce P = { [0, 14)k, 2 Z 14 k nd 3 {0, 7k}, +, n } M is n infinite pseudo subring of M but is not pseudo idel of M. For if T = 0.3k 2.5k 0.5k M nd S = 0.115k 8k 7k P. Clerly T n S = 0.3k 2.5k 0.5k n 0.115k 8k 7k = k 6 13k 7k 0 0 0

87 86 Multidimensionl MOD Plnes = k 8k 3.5k P. Hence P is only pseudo subsemiring nd not n idel of M. Exmple 1.126: Let W = i [0, 27)k; k 2 = 26k, 1 i 9, +, } be MOD non commuttive specil qusi dul number intervl pseudo ring. W hs zero divisors, units nd idempotents. W hs both finite nd infinite order subrings. Infct W hs right idels which re not left idels nd vice vers. However W with n the nturl product is commuttive. Exmple 1.127: Let W = i [0, 40)k, k 2 = 39k, +; n, 1 i 21}

88 Renming of Certin Mthemticl Structures 87 be the MOD specil qusi dul number mtrix intervl pseudo ring. W hs severl subrings of finite order which re not pseudo. W hs severl pseudo subrings of infinite order which re not idels. W hs t lest 21 C C C 20 number of pseudo subrings of infinite order which re idels of W. Study of these hppens to be n interesting problem. However we leve the following s n open conjecture. Conjecture 1.3: Let W = {s t mtrices with entries from [0, m)k, k 2 = (m 1)k, +, n } be the MOD specil qusi dul number intervl pseudo ring of mtrices. i. Cn W hs idempotents where the entries re from [0, m)k \ {0}? ii. Cn W hve S idempotents? Exmple 1.128: Let S = i [0, 15)k, k 2 = 14k, +, n, 1 i 14} be the MOD specil qusi dul number intervl mtrix pseudo ring.

89 88 Multidimensionl MOD Plnes Id = 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k cts s the unit for section of the set S. For tke A = 0.5k 11.2k 5.3k 4.5k 3.1k 5.7k 6.8k 7.5k 9k 8.3k 12k 13.1k 14.2k 14k S. A n Id = 0.5k 11.2k 5.3k 4.5k 3.1k 5.7k 6.8k 7.5k 9k 8.3k 12k 13.1k 14.2k 14k n 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k 14k

90 Renming of Certin Mthemticl Structures 89 = 7 14k k k 3 14k k k k 14 7k 6 14k k 3 14k k k 14k = 8k 5.2k 3.8k 12k 7.6k 7.2k 12.8k 8k 9k 6.8k 12k 2.8k 8.2k 14k A. However Id cts s the identity for finite subring of S given by P = { i Z 15 k, k 2 = 14k, 1 i 14, +, n }. So we cn cll Id s the restricted non pseudo identity of S. Exmple 1.129: Let B = i [0, 17)k, k 2 = 16k, 1 i 12, +, n } be the MOD specil qusi dul number intervl mtrix pseudo ring of infinite order. B hs zero divisors. B hs no unit.

91 90 Multidimensionl MOD Plnes However B hs subrings of finite order which re not pseudo. B hs t lest 12 C C 12 number finite subrings which re not pseudo. Study of these pseudo rings is innovtive. Finding idempotents hppens to be chllenging problem. Severl problems re suggested in the lst chpter of this book.

92 Chpter Two PROPERTIES OF MOD INTERVAL TRANSFORMATION OF VARYING TYPES In this chpter we describe the MOD intervls of different types. In the first plce ny intervl tht hs the cpcity to represent the rel line (, ) will be known s the MOD rel intervl. Infct there re severl such different types of MOD intervls like MOD complex modulo integer intervl, MOD neutrosophic intervl, MOD dul number intervl, MOD specil dul like number intervl nd MOD specil qusi dul number intervl. These intervls for given intervl re infinitely mny. Ech of these will be represented by exmples s well s by the MOD theory behind it. For MOD theory in generl mkes the infinite rel number continuum (, ) into MOD semi open intervl [0, m); 1 m. For we see such MOD rel semi open intervls which represent the rel line (, ) re infinite in number for m tkes ll vlues in Z + \ { }.

93 92 Multidimensionl MOD Plnes Hence we define the notion of MOD rel intervl r trnsformtion. I η : (, ) [0, m). In the first plce MOD rel intervl trnsformtion mps infinite number of points on to singleton point. This is done by the MOD rel intervl trnsformtion periodiclly. This is represented by the following exmple. For instnce [0, 12) be the semi open intervl. r η I (, ) [0, 12) is defined s follows: r η I (12) = 0, η (24) = 0, r I r η I (36) = 0 nd so on. r η I (12.7) = 0.7, r η I (24.7) = 0.7 nd so on. Thus periodiclly these re mpped on fixed point. Likewise r η I (18.9) = 6.9, r η I (6.9) = 6.9, r η I (30.9) = 6.9 nd so on. This is the wy infinite number of periodic points re mpped on to single point.

94 Properties of MOD Intervl 93 Let r η I : (, ) [0, 7) r η I (7) = 0 = r η I (9.8) = 2.8, r η I (14) = r η I (21) = r η I (16.8) = 2.8 nd so on. Throughout our discussions in this book intervl [0, m) 1 < m < is nothing but the MOD intervl of rel intervl (, ). Since it is open intervl we demnd out MOD intervl to be semi open intervl. Further if x (, ) the MOD rel intervl trnsformtion r I η : (, ) [0, m) is defined s r I η (x) = x if 0 < x < m 0 if x = ± m r r if x = s + m m x if x is negtive less thn m m t if x is negtive nd x = n + t m This is the wy MOD intervl trnsformtion is defined. We will just illustrte this sitution by n exmple or two. Exmple 2.1: Let [0, 9) be the MOD intervl.

95 94 Multidimensionl MOD Plnes r I η : (, ) [0, 9) is defined for these number in the following. r I η ( ) = r I η ( ) = r I η ( ) = r I η ( 7.321) = Thus I r η : (, ) [0, 9). It is importnt to note tht in generl there re infinite number of points in (, ) which is mpped onto the intervl [0, 9) s this intervl cn represent (, ) we choose to cll this intervl s MOD intervl. Another dvntge is tht we hve infinitely mny MOD intervls. Infct the intervl [0, 1] fuzzy intervl will be clled s the fuzzy MOD intervl if we tke the semi open intervl [0, 1). The min reson for using [0, m); 1 m <. The min dvntge of using MOD intervl is tht we hve infinitely mny MOD intervls nd s per the need of the problem one cn choose ny of the intervls. This flexibility is not true in cse of (, ) for we hve one nd only one rel line. Further ny entry in (, ) cn be esily nd precisely represented in the MOD intervl [0, m).

96 Properties of MOD Intervl 95 Further ll points in (, ) which re mpped onto the sme point of [0, m) will be clled s eqully represented points or similrly represented points or s points which enjoy the sme MOD sttus. So these points will be MOD similr in the MOD intervl [0, m). We will describe them by n exmple or two. So we hve by this trnsformtion converted infinite number of points to single point by clerly stting these points enjoy the sme MOD sttus or they re similr on the MOD intervl. Such study is not only new but is innovtive; so the infinite line (, ) is mpped to infinite continuum [0, m) without much losing the property of the line. We see ny prcticl problem will hve its solution only to be finite so it is infct wste to work with the infinite line (, ). Further it is pertinent tht most of the vlues which re ccepted re only positive vlues; this lso pertins to prcticl problems. So in this MOD intervls [0, m); 1 m < we see no negtive solution is possible. So we see there re severl dvntges of using these MOD intervls. They re more prcticl, periodic nd fesible prt from sving one from working on very lrge intervl (, ). Further s there re infinite number of MOD intervls one cn choose ny pproprite one which is needed. Certinly use of these MOD intervls cn sve both time nd economy.

97 96 Multidimensionl MOD Plnes Further they re highly recommended for storge becuse of the periodicity they enjoy nd the compctness of these intervls s well s better security. Thus [0, m); these rel MOD intervls is the best lterntive replcement for the rel intervl (, ). The dvntges of using them re: i. There re mny such MOD rel intervls so one hs more choice. ii. Periodic elements which re infinite in number re mpped onto single element. This condenses the intervl very lrgely. iii. Since the negtive prt of the intervl hs no plce in prcticl nd rel world problems we think use of MOD rel intervls will be boon to ny reserchers. iv. Certinly use of MOD intervls in the storge systems will sve certinly time nd money nd give more security. v. It is time chnge should tke plce in the world ruled by MOD technology so these visuliztion or reliztion of the rel intervl (, ) by [0, m) for 1 m < is nothing but MODnizing the intervl nd pproprite m cn be chosen depending on the problem nd its need. We will supply few exmples of them before we proceed onto define other types of MOD intervls. Exmple 2.2: Let [0, 8) be the MOD rel intervl the rel MOD intervl trnsformtion. r I η : (, ) [0, 8)

98 Properties of MOD Intervl 97 r I η (6.8) = 6.8, r I η (16) = 0, r I η (29.3) = 5.3, r I η ( 5.2) = 2.8 nd r I η ( 19.8) = 4.2. Thus ll 8m (m Z) re mpped by intervl. r I η to zero of the MOD Infct [0, 8) is n intervl which represent every element of (, ). However we cnnot sy given x [0, 8) cn we find uniquely the corresponding element in (, ). Infct we cn only give periodic vlues of the corresponding x which re infinite in number. We hve t present no method or mens to fix the preimge of x [0, 8) to unique element in (, ). This is true for ny [0, m); 1 m <. However when one hs ventured to work out with prticulr vlue of m one knows the ctul trnsformtions which re importnt to them. As invribly prcticl problems do not need the negtive prt of the rel line one cn to some extent fix the periodic repeting vlues of (, ) mpped to single vlue in [0, m).

99 98 Multidimensionl MOD Plnes Finlly s the concept of negtive numbers hppens to be very difficult one for the understnding of smll children the replcement of (, ) by [0, m) my be boon to them. Such prdigm of shift cn benefit the eduction system. Exmple 2.3: Let [0, 11) be the MOD rel intervl of (, ) got by the MOD rel intervl trnsformtion. r I η : (, ) [0, 11) by r I η ( 7) = 4, r I η (10) = 10, r I η ( 18) = 4 = r I η ( 7) = r I η (4) = r I η (15) = r I η (26) = r I η ( 29) nd so on. Now hving seen the new notion of MOD rel intervl we now proceed onto define MOD complex intervl. For this intervl ( i, i) is defined s the complex intervl nd i 2 = 1. Now we cll [0, mi F ) to be the complex MOD modulo integer intervl. 2 m < nd i = m 1. Clerly [0, 3i F ) = {i F [0, 3); modulo integer intervl. [0, 6i F ) = {i F [0, 6); modulo integer intervl. 2 F 2 i F = 2} is complex MOD 2 i F = 5} is gin complex MOD We hve infinitely mny such complex MOD modulo integer intervls.

100 Properties of MOD Intervl 99 Now we denote by ( i, i) [0, mi F ); c I η the MOD complex trnsformtion of 2 i F = m 1 nd i 2 = 1. For esy nd simple understnding we will first illustrte this sitution by some exmples. c Exmple 2.4: Let I η : ( i, i) [0, 12i F ) be the MOD complex modulo integer intervl trnsformtion. Define c I η (25i) = i F. Clerly modulo integer intervl trnsformtions. c I η ( 49.8i) = (10.2i F ). c I η (i) = i F in ll MOD complex c I η (12i) = 0, c I η ( 48i) = 0, c I η ( 1.8i) = 10.2i F nd so on. It is clerly seen more thn one point is mpped onto the sme point. Infct we hve infinitely mny such points in ( i, i) which re mpped onto the sme point in [0, mi F ). Thus the MOD complex intervl trnsformtion c I η lso mps periodiclly infinitely mny points in ( i, i) into single point in [0, mi F ). c Exmple 2.5: Let I η : ( i, i) [0, 14i F ) be the MOD complex intervl trnsformtion. c I η (13.8i) = 13.8i F, c I η (24.7) = 10.7i F,

101 100 Multidimensionl MOD Plnes c I η ( 14.3i) = 13.7i F nd c I η ( 27.8i) = 0.2i F. This is the wy MOD complex intervl trnsformtion is crried out. c Exmple 2.6: Let I η : ( i, i) [0, 23i F ) is the MOD complex intervl trnsformtion. Define c I η (42.8i) = 19.8i F c I η (14.384i) = i F. c I η ( 10.75i) = 12.25i F nd so on. Thus MOD complex intervl trnsformtion. Next we will proceed onto define other intervls before we give some properties enjoyed by [0, mi F ). i. The MOD intervl complex modulr trnsformtion mps infinite number of elements in ( i, i) to single elements in [0, mi F ). Infct this occurs periodiclly. However we do not hve unique mens of retrieving the preimges of elements in [0, mi F ). ii. The MOD intervl reduces the problem of leving ( i, 0) nd mking [0, i) onto infinite intervl but smll intervl depending on m. This is compct tht is why we choose to define this s MOD complex modulo integer intervl.

102 Properties of MOD Intervl 101 iii. Another interesting feture bout MOD complex modulr integer intervl is tht we hve infinite number of such intervls like [0, mi F ), 2 m <. iv. The infinite number of MOD complex modulo integer intervls will help in working in better dpttion for n pproprite [0, mi F ) cn be done i = m 1, 2 m <. Next we define the notion of MOD neutrosophic intervls [0, mi) where I 2 = 0; 2 m <. Clerly ( i, i) is the neutrosophic rel intervl. We cll [0, mi) s the MOD neutrosophic intervl where 1 m <. We will first illustrte this sitution by exmples. Exmple 2.7: Let [0, 5I) be the MOD neutrosophic intervl [0, 5I) = {I [0, 5)}. Clerly [0, 5I) is of infinite order yet compct clled s the MOD neutrosophic intervl. Exmple 2.8: Let [0, 7I) be the MOD neutrosophic intervl [0, 7I) = {I [0, 7)}. Clerly [0, 7I) is of infinite order yet compct clled s the MOD neutrosophic intervl. Exmple 2.9: Let [0, 20I) = {I [0, 20)} be the MOD neutrosophic intervl. Now we show how the MOD neutrosophic intervl I trnsformtion I η from ( I, I) to [0, mi); 1 m < is defined. 2 F

103 102 Multidimensionl MOD Plnes I I η (I) = I. First we will illustrte this sitution by some exmples. Exmple 2.10: Let [0, 6I) be the MOD neutrosophic intervl. Define I I η : ( I, I) [0, 6I) s follows: I I η (8I) = 2I, I I η (3.7I) = 3.7I, I I η ( 5I) = I, I I η (27.342I) = 3.342I, I I η ( 14.45I) = 3.55, I I η ( 42.7I) = 5.3I nd so on. This is the wy the MOD neutrosophic intervl trnsformtion is crried out. Exmple 2.11: Let [0, 10I) be the MOD neutrosophic intervl. Define I I η : ( I, I) [0, 10I) s follows: I I η (15.723I) = 5.723I I I η ( 24.62I) = 5.38I I I η (40I) = 0, I I η ( 40I) = 0,

104 Properties of MOD Intervl 103 I I η (250I) = 0, I I η ( 470I) = 0, I I η ( 7.5I) = 2.5I, I I η ( 22I) = 8I nd so on. Exmple 2.12: Let [0, 13I) be the MOD neutrosophic intervl. I Let I η : ( I, I) [0, 13I) define I I η (16.5I) = 3.5I, on. I I η ( 19.7I) = 6.3I, I I η (4.3I) = 4.3I, I I η ( 4.32I) = 8.68I nd so Infct infinite number of points in ( I, I) is mpped onto single point in [0, 13I). Exmple 2.13: Let [0, 4I) be the MOD neutrosophic intervl I I η : ( I, I) [0, 4I) is defined s follows: I I η (10.34I) = 2.34I, I I η ( 10.34I) = 1.66I, I I η (48.3I) = 0.3I, I I η ( 48.3I) = 3.7I nd so on. It cn be esily verified tht infinite number of elements in ( I, I) into single element of [0, 4I). Now hving seen the definitions of MOD neutrosophic intervls nd MOD neutrosophic intervl trnsformtions we now proceed onto enumerte few of its properties ssocited with them. i. The rel neutrosophic intervl ( I, I) is unique but however we hve infinite number of MOD neutrosophic intervls.

105 104 Multidimensionl MOD Plnes ii. However this flexibility of choosing from the infinite number of MOD neutrosophic intervls helps one to choose the pproprite one. iii. Further insted of working with the lrge ( I, I) intervl we cn work with compct smll MOD neutrosophic intervl. iv. This use of MOD neutrosophic intervl [0, mi) 1 < m < cn mke one sve ( I, 0) for negtive numbers re very difficult. v. Infct the MOD neutrosophic intervl trnsformtion mps infinite number of elements into the intervl [0, mi); 1 m <. Next we proceed onto define, describe nd develop the notion of MOD dul number intervls. Consider the dul number intervl ( g, g) where g 2 = 0. The MOD dul number intervl [0, mg) = {g [0, m); g 2 = 0}; 1 m < re infinite in number. We will first illustrte this sitution by some exmples. Exmple 2.14: Let [0, 5g) = {g [0, 5), g 2 = 0} be the MOD dul number intervl. Exmple 2.15: Let [0, 6g) = {g [0, 6); g 2 = 0} be the MOD dul number intervl. Exmple 2.16: Let [0, 12g) = {g [0, 12), g 2 = 0} be the MOD dul number intervl. We will show how the MOD dul number intervl is mpped onto dul number intervl ( g, g) = {g (, ); g 2 = 0}.

106 Properties of MOD Intervl 105 We define the MOD dul number trnsformtion ( g, g) [0, mg) s follows: d I η from d I η : ( g, g) [0, 15g) is defined by d I η (g) = g. d I η (19.6g) = 4.6g, d I η ( 48.37g) = 11.63g, nd so on. so on. d I η (34.6g) = 4.6g, d I η ( 10.7g) = 4.3g, d I η (15g) = 0, d I η (49.6g) = 4.6g, d I η (17.82g) = 2.82g, d I η (30g) = 0, d I η (60g) = 0 d I η (64.6g) = 4.6g nd This is the wy the MOD dul number trnsformtion is defined. Exmple 2.17: Let d I η : ( g, g) [0, 12g). d I η (13.7g) = 1.7g, d I η (20g) = 8g, d I η ( 5g) = 7g, d I η (10.2g) = 10.2g, d I η ( 30.7g) = 5.3g nd so on. Exmple 2.18: Define I d η : ( g, g) [0, 11g) by d I η (121g) = 0, d I η (22g) = 0, d I η ( 11g) = 0, d I η ( 44g) = 0, d I η (66g) = 0, nd so on. d I η (15g) = 4g, d d I η (26g) = 4g, d I η (37g) = 4g Exmple 2.19: Let I η : ( g, g) [0, 8g) be the MOD dul number intervl trnsformtion.

107 106 Multidimensionl MOD Plnes d I η (4g) = 4g, d I η (8g) = 0, d I η ( 6g) = 2g, d I η (14g) = 6g, d I η ( 148g) = 4g, d I η (19g) = 3g nd so on. d Exmple 2.20: Let I η : ( g, g) [0, 53g) be defined by d I η (50g) = 50g, d I η (0.38g) = 0.38g, d I η (6.3g) = 6.3g, d I η ( 14g) = 39g nd so on. d I η (106g) = 0. Thus infinite number of points in ( g, g) is mpped onto single point in the MOD dul number intervl. However for point in [0, mg) we cnnot uniquely get the corresponding point in ( g, g). This is one of the flexibility for we cn choose n pproprite one need s the expected solution of the problem. Now hving seen MOD dul number intervls [0, mg); g 2 = 0; 1 m < we now proceed onto define specil dul like number MOD intervls. We know the specil dul like number intervl is ( h, h) where h 2 = h. Now s in cse of dul numbers we hve infinite number of specil dul like number intervls [0, mh); h 2 = h nd 1 m <. We will first illustrte this sitution by some exmples. Exmple 2.21: Let [0, 8h); h 2 = h be the MOD specil dul like number intervl.

108 Properties of MOD Intervl 107 Exmple 2.22: Let [0, 12h); h 2 = h be the specil MOD dul like number intervl. Exmple 2.23: Let [0, 18h); h 2 = h be the specil dul like MOD number intervl. Infct we hve infinitely mny MOD specil dul like number intervls but only one specil dul like number rel intervl ( h, h); h 2 = h. Now s in cse of other intervls we in cse of MOD specil dul like number intervls lso obtin MOD specil dul like number MOD intervl trnsformtion. dl I η : ( h, h) [0, mh); h 2 = h; here dl I η (h) = h. We will first illustrte this sitution by some exmples. Exmple 2.24: Let [0, 9h); h 2 = h be the MOD specil dul like intervl. dl I η : ( h, h) [0, 9h) be defined s dl I η (8.3h) = 8.3h, dl I η (18.34h) = 0.34h, dl I η (9.34h) = 0.34h, dl I η (h) = h. dl I η ( 8.66h) = 0.34h nd so on. Infct infinitely mny points in ( h, h) re mpped onto single point in [0, mh), h 2 = h; 1 m <. Exmple 2.25: Let [0, 19h); h 2 = h be the MOD specil dul like number plne. I : ( h, h) [0, 19h) is defined by dl I η (25h) = 6h dl η

109 108 Multidimensionl MOD Plnes dl I η (25h) = 6h dl I η ( 10h) = 9h, dl I η (17h) = 17h nd so on. Thus periodiclly infinite number of points re mpped onto the single point for instnce (n )h for vrying < n < is mpped by I ((n )h) = 0.558h. dl η Thus these points (19n )h for vrying n; < n < re clled s periodic points. These periodic points re infinite nd re mpped to 0.558h. Likewise every point in ( h, h) is point which is periodic. Infct if k is deciml lying in (0, 1) then ll (nm + k)h ( h, h) for n Z re mpped by the MOD intervl specil dul like number trnsformtion into k. We cll the collection (nm + k)h ( h, h) for fixed k in (0, 1) s the infinite periodic elements of ( h, h) which re mpped by I into k [0, mh). dl η Since the number of k s in [0, 1) is infinite so we hve infinite number of periodic elements which re mpped into single element in [0, mh). Thus t every stge we wish to cll these MOD specil dul like intervls s they re smll but non bounded in comprison with ( h, h). Next we proceed onto the specil qusi dul number rel intervl ( k, k); k 2 = k. We see [0, mk); k 2 = (m 1)k. 1 m < ; is the MOD specil qusi dul number.

110 Properties of MOD Intervl 109 Define I dq η : ( k, k) [0, mk) by dq I η (k) = k, dq I η (mt.k) = 0 for ll t Z. There re infinite number of elements in ( k, k) which is mpped onto the zero element of [0, mk). dq I η ((mt + r)k) = rk [0, mk); r (0, m). Consider [0, 6k); k 2 = 5k. Define dq I η (15.032k) = 3.032k [0, 6k). dq I η ( 4.5k) = 1.5k, dq I η (27.05k) = 3.05k, dq I η (7.5k) = 1.5k, dq I η (13.5k) = 1.5k, dq I η (25.5k) = 1.5k, dq I η (43.5k) = 1.5k nd so on. Let dq I η : ( k, k) [0, 23k) functions s follows: dq I η (40.3k) = 17.3k; dq I η ( 3k) = 20k; dq I η (43k) = 20k, dq I η (66k) = 20k, we see severl elements in ( k, k) onto single element of [0, 23k).

111 110 Multidimensionl MOD Plnes dq I η Thus ll specil qusi dul number intervl trnsformtion mps periodiclly infinite number of elements into single element in [0, mk). Further we cnnot s in cse of other elements mp elements in [0, mk) into single element in ( k, k). The reverse trnsformtion is not tht esy. Let [0, 10k); k 2 = 9k be the MOD specil qusi dul number intervl. Let x = 8.009k ( k, k) we see dq I η = 8.009k. dq I η (48.64k) = 8.64k, dq I η (0.98k) = 0.98k, dq I η ( 9.2k) = 0.8k, dq I η (12.53k) = 2.53k, dq I η ( 43.42k) = 6.58k nd so on. This is the wy the MOD specil qusi intervl trnsformtion is crried out. Exmple 2.26: Let [0, 12k); k 2 = 11k be the MOD specil qusi dul number intervl. dq I η : ( k, k) [0, 12k) is defined s follows: dq I η (49.372k) = 1.372k, dq I η (10k) = 10k, dq I η ( 7k) = 5k nd dq I η ( 243.7k) = 8.3k.

112 Properties of MOD Intervl 111 This is the wy the MOD specil qusi intervl trnsformtion is crried out. Infct we hve infinitely mny periodic elements in ( k, k) mpped onto [0, 12k). However the mpping of dq T η : [0, 12k) ( k, k) is n one to one embedding which we cll s the trivil mp. If we mp dq T η (x) = (12n + x)k for every x [0, 12k) nd n (, ) then certinly it is not the usul mp for in no plce n element from domin spce is mpped onto infinite number of elements in ( k, k) we cn this s specil pseudo mp. For if [0, 5k); k 2 intervl then t (, ). dq T η = 4k is the specil qusi dul number (x) = 5tk + x this is true for ll x [0, 5k); For instnce x = 4.5k is in [0, 5k) then dq T η (4.5k) = (5t + 4.5)k t (, ). So for the first time we define such mp. We see dq I η dq T η = dq T η o dq I η = I. For instnce if insted η: (, ) [0, m) then η : [0, m) (, ) such tht every x [0, m) is mpped T onto mt + x, t (, ). η T η = η η = identity mp. T Similrly for η : ( I, I) [0, mi) we hve T I such tht I T I : [0, mi) ( I, I) with

113 112 Multidimensionl MOD Plnes T I (x) = (mt + x); m (, ) so x is mpped onto infinite number of points. on. However T I is not mp. Similrly for η c we hve T c nd so These re not mps but specil type of pseudo functions which mps single point in [0, mi F ) into infinitely mny elements in ( i, i).

114 Chpter Three INTRODUCTION TO MOD PLANES In this chpter we for the ske of completeness recll the new notion of MOD plnes introduced in [24]. This is done for two resons (i) s the subject is very new nd (ii) if they re reclled the reder cn redily know those concepts without ny difficulty. We just recll them in the following. DEFINITION 3.1: Let R n (m) = {(, b), b [0, m)}; we define R n (m) s the MOD rel plne built or using the intervl [0, m). We see the whole of the rel plne R R cn be mpped onto the rel MOD plne R n (m). This is done by the trnsformtion η: R R R n (m).

115 114 Multidimensionl MOD Plnes by η(, b) = (0, 0) if = ms nd b = mt b t (0, t) if = ms nd = v + m m u (u, 0) if = s + nd b = mr m m r b s (r,s) if = t + nd = u + m m m m (,b) if < m nd b < m The trnsformtion η is such tht severl pir in R or infinite number of pirs in R re mpped onto single point in R n (m). We hve clled this s rel MOD trnsformtion of R R to R n (m). We will illustrte this sitution by some exmples. Exmple 3.1: Let R n (3) = {(, b), b [0,3)} be the MOD plne on the MOD intervl [0,3). Clerly if (23.7, 84.52) R R then by the MOD trnsformtion η : R R n (3). We get η(23.7, 84.52) = (2.7, 0.52) R n (3). If x = ( 5.37, 2.103) R R then η(x) = {(0.63, 2.103)}. This is the wy MOD trnsformtion from the rel plne R R to the MOD plne R n (3) is crried out. The MOD rel plne on the MOD intervl [0, 3) is s given Figure 3.1. Exmple 3.2: Let R n (8) = {(, b), b [0, 8)} be the rel or Euclid MOD plne on the MOD intervl [0, 8).

116 Introduction to MOD Plnes 115 y 2 (1,1.23) Figure 3.1 x 8 y (2,4.45) x Figure 3.2

117 116 Multidimensionl MOD Plnes Let x = (27.003, 4.092) R R. Now the MOD trnsformtion of x is s follows. η(x) = (3.003, 4.092) R n (8). We give the MOD plne R n (8) in Figure 3.2. Exmple 3.3: Let R n (13) be the rel MOD plne on the MOD intervl [0, 13). Let us define η : R R R n (13) by η((18.021, )) = (5.021, ) R n (13). The MOD Euclid plne is s follows: 13 y (5.021,3.0079) Figure 3.3 x Exmple 3.4: Let R n (10) be the MOD rel plne. Let η be mp from R R R n (10).

118 Introduction to MOD Plnes 117 Tke x = (98.503, 5.082) R R. η(x) = (8.503, 4.918) R n (10). η is MOD rel trnsformtion. Now for properties of MOD rel or euclid plnes refer [24]. Next we proceed onto recll the definition of the notion of MOD complex plne nd the MOD complex trnsformtion from C to C n (m) defined by η c : C C n (m) where 2 C n (m) = { + bi F = (, b) i F = m 1,, b [0, m)} is the complex modulo integer MOD plne defined in [24]. Exmples if them will be provided before the complex MOD trnsformtion is defined. Exmple 3.5: Let C n (15) = { + bi F = (, b), MOD complex modulo integer plne. 2 i F = 14} be the Exmple 3.6: Let C n (13) = { + bi F = (, b); nother complex modulo MOD integer plne. 2 i F = 12} be Exmple 3.7: Let C n (48) = { + bi F = (, b); complex modulo integer MOD plne. 2 i F = 147} be It is importnt to keep on record tht we hve infinite number of complex modulo integer plnes, just like we hve infinite number of rel or euclid MOD integer plnes. Now we just give the digrmmtic representtion of them. Exmple 3.8: Let C n (6) be the complex modulo MOD integer plne given by Figure 3.4. Exmple 3.9: Let C n (11) = { + bi F = (, b); complex finite modulo integer MOD plne. This is given in Figure i F = 10} be the

119 118 Multidimensionl MOD Plnes 6 if (1.5,2) if Figure 3.4 x Figure 3.5 x

120 Introduction to MOD Plnes 119 Exmple 3.10: Let C n (9) = { + bi F = (, b); complex finite modulo integer MOD plne. The grph or the MOD plne C n (9) is s follows: 2 i F = 8} be the 9 if (2.4,5) Figure 3.6 x Now we give few illustrtions of MOD complex modulo integer trnsformtion. Let C n (16) = { + bi F = (, b); i 2 F = 15}. Let x + iy = i C. η c (i) = i F is lwys fixed.

121 120 Multidimensionl MOD Plnes η c (x + iy) = (7.5, 3.312i F ) = i F C n (16). This is the wy the MOD complex trnsformtion is crried out. Let x 1 + iy 1 = i F C. η c (x 1 + iy 1 ) = i F C n (16). Thus both x + iy nd x 1 + iy 1 C re mpped onto the sme point in C n (16); however both x + iy nd x 1 + iy 1 re distinct. Let C n (7) = { + bi F = (, b); modulo integer MOD plne. 2 i F = 6} be the complex The plne is represented in the following: 7 if Figure 3.7 x

122 Introduction to MOD Plnes 121 Let ( , 17.5i) C tht is z = i F η c (z) = (1.669, 3.5i F ) C n (7). Let z 1 = x 1 + iy 1 = η c (z 1 ) = (0.0092, 0.003). This is the wy the MOD complex trnsformtion is crried out. Let z 2 = x 2 + iy 2 = (x 2, y 2 ); ( , ) C. η c (z 2 ) = (5.0032, 5.688) = i F C n (7). Next we give some exmples of some more trnsformtion. Let C n (20) = { + bi F, b [0, 20); complex finite modulo integer plne. 2 i F = 19} be the MOD We show how MOD complex trnsformtion η c from C to C n (20) is defined η c : C C n (20) defined in the wy. Let z = i C η c (z) = (0, 0). Let z 1 = i C; η c (z 1 ) = (12, 10) = i F. Let z 2 = i C. η c (z 2 ) = (12, 5.32) = i F Let z 3 = i C. η c (z 3 ) = (16.25, ) = i F C n (20). Let z 4 = i C. η c (z 4 ) = i F C n (20). This is the wy the MOD complex trnsformtion tkes plce.

123 122 Multidimensionl MOD Plnes Now for more bout MOD complex plne nd MOD complex trnsformtion refer [24-5]. Next we proceed onto recll the notion of neutrosophic MOD plnes. I R n (m) = { + bi, b R, I 2 = I,, b [0, m)} is the neutrosophic MOD plne built using the intervl [0,m). For more bout this refer [24-5]. However for the ske of self continment few exmples of neutrosophic MOD plnes nd the MOD trnsformtion will be provided for the reder. I Exmple 3.11: Let R n (5) = { + bi, b [0, 5), I 2 = I} is the MOD neutrosophic plne built on the MOD intervl [0, 5). 5 I 4 3 (3.1,2.7) Figure 3.8 x

124 Introduction to MOD Plnes 123 If x = I then x in the bove plne is mrked. I Exmple 3.12: Let R n (10) = { + bi, b [0, 10), I 2 = I} be the MOD rel neutrosophic plne built on [0, 10). The grphic representtion is s follows. We plot x = I, y = I nd z = I in this MOD plne. 10 I (5.7,3) (6.8,4.6) (9.3,2.7) 1 (5.7,0) (6.8,0) (9.3,0) Figure 3.9 x Exmple 3.13: Let I R n (13) = { + bi, b [0, 13), I 2 = I} be the MOD neutrosophic plne built on [0, 13).

125 124 Multidimensionl MOD Plnes Let us plce the point x = (5, 5I) = 5 + 5I, y = I nd z = I in Figure Now hving defined nd illustrted by exmples of MOD neutrosophic plnes we just show how the MOD neutrosophic trnsformtion is performed. Exmple 3.14: Let I R n (10) = { + bi, b [0, 10), I 2 = I} be the MOD neutrosophic plne. I (0,5) (5,5) 5 (0,4.2) (2.7,4.2) 4 (0,3.1) (10.5,3.1) (2.7,0) (5,0) (10.5,0) Figure 3.10 x

126 Introduction to MOD Plnes 125 Let η I : R I I R n (10) be defined s follows. Let Let x = I R I η I (x) = I = (7.33, 3.42) R (10). y = I R I I n η I (y) = I R (10) η I is MOD neutrosophic trnsformtion of R I to I R n (10). η I (I) = I lwys. Consider z = I R I η I (z) = I R (10). This is the wy opertion is performed. Thus it is cler z nd y in the neutrosophic plne R I re distinct but however I z nd y re identicl in the MOD neutrosophic plne R n (10). Thus infinite number of points in R I re mpped onto single point in the MOD neutrosophic plne R (10). Thus η is mny to one trnsformtion from R I to I n I n I n I R n (10). However the entire neutrosophic rel plne is mpped onto the one qudrnt but infinite compct neutrosophic MOD plne. Exmple 3.15: Let R (7) = { + bi, b [0, 7), I 2 = I} I n be the MODneutrosophic plne on the MOD intervl [0, 7). Let x = I R I.

127 126 Multidimensionl MOD Plnes Using the MOD trnsformtion η I : R I I R n (7) η I ( I) = I R (7) I n Let Let Let y = I R I ; η I (y) = ( I) I R n (7). z = I R I. η I (z) = I R (7) v = I R I. η I (v) = I R (7) I n I n This is the wy the MOD neutrosophic trnsformtion re I performed from R I R n (7). I Exmple 3.16: Let R n (6) be the MOD neutrosophic plne on the MOD intervl [0, 6). Let x = I R I ; η I : R I I R n (6) η I (12.7 +_ 30.5I) = I R (6). Thus this MOD neutrosophic trnsformtion cn mp nd I will mp infinite number of elements in R I in R n (6). I n For y = I R I is mpped by η I to I. Let z = I R I

128 Introduction to MOD Plnes 127 is mpped by η I to I nd so on. Thus this MOD neutrosophic trnsformtion η I mps infinite I number of coordintes in R I onto the MOD plne R n (m) built on the MOD intervl [0, m). Thus this property will be exploited. Further the dvntge of using MOD neutrosophic plne is tht insted of working on the four qudrnt of the usul rel neutrosophic plne R I ; we cn work on the single compct qudrnt yet infinite. The following figure suggest why the concept of MOD is dvntgeous over R I. 7 I x (-1,-1) Figure 3.11

129 128 Multidimensionl MOD Plnes Figure 3.12 Now hving seen the notion of MOD neutrosophic plne we proceed onto describe nd develop MOD dul number plne nd their generliztions. We will illustrte this sitution by some exmples. Exmple 3.17: Let R n (5)(g) = { + bg, b [0, 5) g 2 = 0} be the MOD dul number plne built using the intervl [0, 5). R n (5)g = {(, b) = + bg} is lso nother wy of representing MOD dul number plne. The dvntge of using MOD dul number plne is insted of working with four qudrnt we work compctly on only one qudrnt. This is illustrted by the following grphs.

130 Introduction to MOD Plnes 129 R(g) = { + bg = (, b), b R, g 2 = 0} is the rel dul number plne which hs four qudrnts. The work in this direction cn be hd from [24]. Further it is pertinent to keep on record tht we cn develop this dul number plne esily to n dimensionl dul number plne. However t this juncture we use only two dimension plne. 9 g (-1,-1) -2 x -3 Figure 3.13

131 130 Multidimensionl MOD Plnes Figure 3.14 Exmple 3.18: Let R n (8)(g) = { + bg = (, b), b [0, 8), g 2 = 0} be the MOD dul number plne built on the intervl [0, 8). 8 g (0,3) (4.8,3) (4.8,0) Figure 3.15 x

132 Introduction to MOD Plnes 131 Next we proceed onto describe nd develop these MOD dul number plnes nd the MOD dul trnsformtions from R(g) to R n (m)(g). Exmple 3.19: Let R n (9)g be the dul number MOD plne. The grph of the plne is given in Figure If x = g then it is plotted s bove. Exmple 3.20: Let R n (6)g be the MOD dul number plne built using [0, 6). The plne of R n (6)g nd points in R n (6)g re given in the Figure g 8 7 (0,6) (5.5,6) (5.5,0) Figure 3.16 x

133 132 Multidimensionl MOD Plnes 6 g (0,5) (5.1,5) 5 (0,4.8) (2.5,4.8) (0,1) (2,1) 1 (2,0) (5.1,0) (2.5,0) Figure 3.17 x Let x 1 = g, x 2 = g, x 3 = g + 2 R n (16). The grph of them is s bove. Now hving seen how the representtion in the MOD dul number plne looks we now proceed onto describe the MOD dul number trnsformtions η from R(g) to R n (m)g in the following. Recll R(g) = { + bg, b R, g 2 = 0} nd R n (m)g = { + bg, b [0, m), g 2 = 0}. Let x = g R(g) to find the point x in R n (9)g. η g (g) = g nd η g (g) = η g (x) = η g ( g) = (4 + 6g)

134 Introduction to MOD Plnes 133 = (4, 6) R n (g). Let x 1 = g R(g) η g (x 1 ) = 0. Let x 2 = g R(g) η g (x 2 ) = (8, 2) = 8 + 2g. Let x 3 = g R(g) η g (x 3 ) = (4.77, 8.34) = g R n (g). Now hving seen the exmples of MOD dul number plnes nd the MOD dul number trnsformtion we proceed onto describe the MOD specil dul like number plnes. For more bout this concept plese refer [24-5]. 2 R n (m)(h) or R n (m)(g 1 ) = { + bg 1, b [0, m), g 1 = g 1 } is defined s the specil MOD dul like number plne [17]. Infct with vrying m; 1 < m < we hve infinitely mny such plnes. We will illustrte this by some exmples. Exmple 3.21: Let R n (10)g 1 = { + bg 1 = (, b) g = g 1,, b [0, 10)} be the MOD specil dul like number plne. Let y = g 1 nd x = g 1 R n (10)g. The grph of its given below: 2 1

135 134 Multidimensionl MOD Plnes 10 g1 9 8 (0,7) (3.5,7) (0,2.7) 3 (9.6,2.7) 2 1 (3.5,0) (9.6,0) x Figure 3.18 Exmple 3.22: Let 2 R n (11)g 1 = {(, b) = + bg 1, b [0, 11), g 1 = g 1 } be the specil dul like number MOD plne on the MOD intervl [0, 11). Let nd x = g 1, y = g 1 z = g 1 R n (11)g 1. This grph is given below:

136 Introduction to MOD Plnes g (0,5) (1.8,5) 5 (0,3.4) 4 (7.5,3.4) (0,2.9) (9.5,2.9) (1.8,0) (7.5,0) (9.5,0) x Figure 3.19 Exmple 3.23: Let R n (12)g 1 = {(, b) = + bg 1, b [0, 12), be the specil MOD dul like number plne. 2 g 1 = g 1 } We see we cn fix or plot the given points on this plne. Let x = g 1, nd y = 8 + 6g 1 z = g 1 R n (12)g 1.

137 136 Multidimensionl MOD Plnes The grph is s follows: 12 g (0,7.5) 8 (10.2,7.5) 7 (0,6) (8,6) (0,2) (4.3,2) 2 1 (4.3,0) (8,0) (10.2,0) x Figure 3.20 Next we proceed onto describe the MOD specil dul like number trnsformtion. η g 1 : R(g 1 ) R n (m) g 1. η g 1 (g 1 ) = g 1 for ny + bg 1 = x η g 1 (x) = (d, u). Here = c + d nd b = t + u m m m m

138 Introduction to MOD Plnes 137 We will illustrte the vrious vlues of x = + bg 1 in R(g 1 ). Let x = g 1 R(g 1 ) to find the specil qusi dul MOD trnsformtion η : R(g 1 ) R n (10)g 1 ; g 1 Let η g 1 (x) = g 1 R n (10)g 1 y = g 1 R(g 1 ) Let Let η g 1 (y) = g 1 R n (10)g 1. z = g 1 R(g 1 ) η g 1 (z) = g 1 R n (10)g 1. u = g 1 R(g 1 ) η g 1 (u) = g 1 R n (10)g 1. Let v = g 1 R(g 1 ) η g 1 (v) = g 1 R n (10)g 1. We see x, z nd v in R(g 1 ) re mpped onto the sme point g 1 Let x = g 1 R(g 1 ) now to find where x is mpped in the MOD specil dul like number plne R n (5)g 1. Let η g 1 (x) = (4.72, 4.331) = g 1 R n (5)g 1. y = g 1 R(g 1 ) η g 1 (y) = g 1 R n (5)g 1.

139 138 Multidimensionl MOD Plnes We see x y in R(g 1 ) but η g 1 (x) = η g 1 (y). Thus we see severl points cn be mpped onto single point. Infct we hve infinite number of points in the specil dul like number plne R(g 1 ) mpped onto single point. Thus the entire specil dul like number plne R(g 1 ) is mpped onto the MOD specil dul like number plne R n (5)g 1 tht is the ll elements from ll the four qudrnts of R(g 1 ) is mpped onto the R n (5)g 1 which is only single qudrnt MOD plne. Thus this MOD plne is not only considered compct but one hs choice to choose ny number nd get the plne without leving single point in the specil dul like number plne R(g 1 ). Further we hve infinite number of such MOD specil dul like number plnes. Let us consider the MOD specil dul like number plne R n (12)g 1. Now we show how R(g 1 ) is trnsformed into R n (12)g 1 using MOD specil dul like number trnsformtions η g 1 : R(g 1 ) R n (12)g 1. Let us consider the element x = g 1 R(g 1 ). η (g 1 ) = g 1 nd η (x) = g 1 R n (12)g 1 g 1 η g 1 (y) = g 1 η g 1 ( g 1 ) = 1.4 R n (12)g 1 η g 1 ( g 1 ) = ( ) = 3.3 R n (12)g 1.

140 Introduction to MOD Plnes 139 η g 1 ( g 1 ) = 0 R n (12)g 1. η g 1 ( g 1 ) = g 1 R n (12)g 1. This is the wy MOD specil dul like number trnsformtion is crried out from R(g 1 ) onto R n (12)(g 1 ). Next we consider the specil qusi dul number MOD plnes using the specil qusi dul numbers. 2 Recll R(g 2 ) = { + bg 2, b R; g 2 = g 2 } is defined s the rel specil qusi dul number plne. 2 However R(g 2 ) = { + bg 2 = (, b), b R; g 2 = g 2 }. Now we proceed onto give the notion of specil MOD qusi dul number plnes. 2 R n (m)g 2 = { + bg 2, b [0, m), g 2 = (m 1)g 2 } = {(, b) 2, b [0, m); g 2 = (m 1)g 2 } is defined s the specil MOD qusi dul number plne defined on the MOD intervl [0, m). We will first illustrte this sitution by some exmples. Exmple 3.24: Let R n (10)g 2 = { + bg 2 = (, b) g = 9g 2,, b [0, 10)} be the MOD specil qusi dul number plne built on the MOD intervl [0, 10). Exmple 3.25: Let R n (7)g 2 = { + bg 2 = (, b), b [0, 7); g 2 = 6g 2 } is the MOD specil qusi dul number plne built on the MOD intervl [0, 7).

141 140 Multidimensionl MOD Plnes 7 g (0,3.5) (2.4,3.5) (2.4,0) x Figure 3.21 Exmple 3.26: Let R n (12)g 2 = { + bg 2 = (, b), b [0, 12); 2 g 2 = 11g 2 } be the specil MOD qusi dul number plne using the intervl [0, 12). Let x = g 2 nd y = g 2 R n (12)g 2. The grph of them is s follows:

142 Introduction to MOD Plnes g2 (0,11) (5.5,11) (0,3.2) (10,3.2) (5.5,0) (10,0) x Figure 3.22 Exmple 3.27: Let 2 R n (18)g 2 = { + bg 2 = (, b), b [0, 18); g 2 = 17g 2 } be the MOD specil qusi dul number plne built on the intervl [0, 18). We next proceed onto show how the MOD specil qusi dul number trnsformtion from R(g 2 ) to R n (m)g 2 is performed. Let us tke m = 18. Supposeη : R(g 2 ) R n (m)g 2 we mp g 2 Now if x = g 2 then η g 2 (g 2 ) = g 2. η g 2 (x) = g 2 R n (m)g 2.

143 142 Multidimensionl MOD Plnes Let y = g 2 R(g 2 ) η g 2 (y) = g 2 R n (m)g 2. Next let z = g 2 R(g 2 ) η g 2 (z) = g 2 R n (m)g 2. Clerly x, y, z R(g 2 ) re ll distinct, however η (x) = η (y) = η (z) = g 2. g 2 g 2 g 2 Thus η the MOD specil qusi dul number trnsformtion mps severl elements onto single element, infctη mps infinite number of elements in R(g 2 ) onto single element in R n (m)g 2. This is one of the vitl properties enjoyed by MOD trnsformtions in generl nd MOD qusi specil dul numbers trnsformtion in prticulr. Now hving seen ll these plnes nd their structure we now proceed onto define multi dimensionl MOD plnes of very mny type. g 2

144 Chpter Four HIGHER AND MIXED MULTIDIMENSIONAL MOD PLANES In this chpter we introduce the notion higher dimensionl MOD plnes of different types. We hve 6 types of different higher or multidimensionl plnes prt from the infinite number of mixed multidimensionl plnes. We will define, describe nd develop these new multi structure plnes in this chpter. We know R n = {R R} is n-dimensionl rel spce; R n = {( 1, 2,, n ) i R; 1 i n}. R n is commuttive ring with respect to + nd. Similrly C n = {(C 1, C 2,, C n ) C i C the complex plne i = 1, 2, 3,, n} is the multi or n dimensionl complex plne.

145 144 Multidimensionl MOD Plnes x = {(5 + 2i, 3 + 4i,, i)} be ny specific element in C n with usul + nd ; C n is lso ring clled the multidimensionl complex ring or spce. Likewise ( R I ) m = {(b 1,, b m ) b i (R I) = { + bi, b R; I 2 = I} 1 i m} m N is the multidimensionl or m dimensionl neutrosophic spce. Clerly ( R I ) n under usul + nd is ring. Next [R(g)] p = {( 1 + b 1 g,, p + b p g) where i, b i R; 1 i p, g 2 = 0} is defined s the p-dimensionl or multidimensionl dul number spce; p N. [R(g)] p is ring under usul + nd. Now (R(h)) t = {( 1 + b 1 h, 2 + b 2 h,, t + b t h) i, b i R; 1 i t; h 2 = h} is defined s the t-dimensionl or multidimensionl specil dul like number spce, t N. Clerly (R(h)) t under + nd is ring. (R(k)) m = {( 1 + b 1 k, 2 + b 2 k,, m + b m k) i, b i R; 1 i m, k 2 = (m 1)k} is multidimensionl specil qusi dul number spce. (R(k)) m is ring with respect to + nd. We will first illustrte this sitution by some exmples. Exmple 4.1: Let R 5 = {( 1, 2,, 5 ) where i R; 1 i 5, +, } be ring which is multidimensionl spce. x = (0.5, 3, 5.2, 7, 8) nd

146 Higher nd Mixed Multidimensionl MOD Plnes 145 y = (43, 10.8, 11, 17, 4.331) re elements of R. x + y nd x y R 5 it is not difficult to verify. Exmple 4.2: Let C 7 = {( 1, 2,, 7 ) i C; 1 i 7, +, } be ring. x = (0.5i, 8 + 3i, 4.5i, 7, 0, 3i, 10 4i) nd y = (0, 3i, 2, 4 2i, i, 10 2i, 0) C 7 x + y = (0.5i, 8 + 6i, 6 5i, 11 2i, i, 10 + i, 10 4i) nd x y = (0, 24i 9, 8 10i, 28 14i, 30i + 6, 0) C 7. This is the wy opertions re performed on C 7. Exmple 4.3: Let R I 4 = {( 1, 2, 3, 4 ) i R I; 1 i 4, +, } be the multidimensionl neutrosophic ring. Let x = (5 + 2I, 7I, 4, 10 I) nd y = (0, 8 4I, I, 4I) ( R I ) 4. x + y = (5 + 2I, 8 11I, 7 + 5I, 10 5I), x y = (0, 28I, I, 36I) R I. This is the wy opertions re performed on R I 4.

147 146 Multidimensionl MOD Plnes Exmple 4.4: Let (R(g)) 3 = {( 1, 2, 3 ) i R(g), 1 i 3, +, } be the multidimensionl dul number ring. Let x = (5 3g, 2 + g, 0) nd y = (4 + 2g, 5 g, 7 + 3g) (R(g)) 3. x + y = (9 g, 3, 7 + 3g). x y = (20 2g, g, 0) (R(g)) 3. This is the wy opertions re performed. Exmple 4.5: Let [R(h)] 5 = {( 1, 2,, 6 ) i R(h), h 2 = h; 1 i 6, +, } be multidimensionl specil dul like number ring. Let x = (0, 3 + h, 0, 4 2h, 0, 6 + h) nd y = (7 + 8h, 0, 6 + 5h, 3h, 8 + 4h, 2) (R(h)) 5. x + y = (7 + 8h, 3 + h, 6 + 5h, 4 + h, 8 + 4h, 4 + h) nd x y = (0, 0, 0, 6h, 0, h) (R(h)) 6. This is the wy the opertions + nd re crried out in (R(h)) 6. Exmple 4.6: Let (R(k)) 4 = {(b 1, b 2, b 3, b 4 ) b i R(k); 1 i 4, k 2 = k, +, }

148 Higher nd Mixed Multidimensionl MOD Plnes 147 be the multidimensionl specil qusi dul number plne. Let x = (7 k, 8 + 3k, 4k, 5 4k) nd y = (8k, 4 + 2k, 8 + 5k, 3k) (R(k)) 4. x + y = (7 + 7k, k, 8 + 9k, 5 k) nd x y = (64k, k, 12k, 27k) (R(k)) 4. This is the wy opertions re performed on (R(k)) 4. Next we proceed onto define mixed multidimensionl spces. We will first illustrte this sitution by some exmples. Exmple 4.7: Let R C R I C R(g) = {(, b, c, d, e) R 2, b, d C, c R I nd e R(g); g 2 = 0, I 2 = I, i = 1} = M; M is defined s the mixed multidimensionl spce. We see how + nd opertions re performed on M. Let x = (5, 3 4i, I, 2 + I, 3 + 4g) nd y = ( 2, 6 + 3i, 3 + I, 4i, 2 g) M. x + y = (3, 9 i, 7 + 5I, 2 + 5i, 1 + 3g) M x y = ( 10, 30 15i, I, 8i 4, 6 12g) M. This is the wy opertions re performed on the mixed multidimensionl spces.

149 148 Multidimensionl MOD Plnes Exmple 4.8: Let S = C C C R(g) R(g) R(k) R I = {( 1, 2, 3, 5, 6, 7 ) i C; 1 i 3, 4, 5 R(g); g 2 = 0, 6 R(k) nd 7 R I, I 2 = I, k 2 = k, +, } be the mixed multi dimensionl spce. Now if x = ( 3 + 2i, 7 I, 4 I, 2g + 1, 5g, 10k 4, 8 + 2I) nd y = (6 i, 2i, 5, 4g, 8g, 3k, 4) S; x + y = (3 + i, 7 + i, 1 i, 6g + 1, 9g, 13k 4, I) nd x y = ( 16i + 15i, 14i + 2, 20 5i, 4g, 0, 42k, I) S. This is the wy opertions re crried out on S. Exmple 4.9: Let N = R R R(k) R(k) R(h) R(h) R(g) C = {(, b, c, d, e, f, g, h), b R, c, d R(k), k 2 = k, e, f R(h), h 2 = h, g 1 R(g) g 2 = 0 nd h C} be the mixed multidimensionl spce. Now if x = (5, 2, 3 + 8k, 4 k, h 3, 4h + 1, 8g, 10 i) nd y = ( 7, 8, 5k, 3 + 2k, h, 4h + 1, g, 10 + i) N. x + y = ( 2, 6, k, 7 + k, 2h 3, 8h + 2, 9g, 20); x y = ( 35, 16, 25k, k, 2h, h, 101) N.

150 Higher nd Mixed Multidimensionl MOD Plnes 149 This is the wy sum nd product opertions re performed on the mixed multidimensionl spces. Exmple 4.10: Let B = {C R R(g) C R(h) R C} = {( 1, 2, 3, 4, 5, 6, 7 ) 1, 4, 7 C, 2, 6 R, 3 R(g); g 2 = 0, 5 R(h); h 2 = h, + } be the mixed multidimensionl spce. Clerly for x = (10 + i, 7, 8g + 2, 3i, 5 + h, 3, 2 + 4i) nd y = (3, 1, 5g, 6 + i, h, 6, 0) B. x + y = (13 + i, 6, 13g + 2, 6 + 4i, 5 + 2h, 9, 2 + 4i) nd x y = (30 + 3i, 7, 10g, 18i 3, 6h, 18, 0) re in B. This is the wy opertions re performed. Now we cn just define mixed multidimensionl spce. DEFINITION 4.1: Let M = {P 1 P 2 P n } where P i {C, R, R(g), g 2 = 0; R(h), h 2 = h; R(k); k 2 = k, R I }; 1 i n be the mixed multidimensionl spce. M is ring under usul + nd. Now it is importnt to note tht ll these mixed multidimensionl spces re vector spce of multidimension over the rels R. However they re in generl not mixed multivector spces over R I, R(g), g 2 = 0; R(h); h 2 = h; R(k) = k 2 = k nd so on. Exmple 4.11: Let M = {(R R(g) C R(h) C) g 2 = 0, h 2 = h; +}

151 150 Multidimensionl MOD Plnes be the mixed multidimensionl vector spce over R; however M is not mixed multidimensionl vector spces over R(g) or R(h) or C. Exmple 4.12: Let M = {(C C C R I R I R(g) R(h)) g 2 = 0, h 2 = h, I 2 = I, +} be mixed multidimensionl vector spce over R. M is not mixed multidimensionl vector spce over C or R(g) or R(h) or R I. Exmple 4.13: Let V = {(R(g) R(h) R(k) R(g) R I C C R(h)) g 2 = 0, h 2 = h, I 2 = I, k 2 = k, i 2 = 1, +} be mixed multi dimensionl vector spce only over R. Clerly V is not mixed multidimensionl vector spce over R(h) or R(g) or C or R I or R(k). Now we proceed onto define, describe nd develop multidimensionl MOD plnes nd MOD mixed multidimensionl plnes. First we will describe them by exmples. Exmple 4.14: Let M = [0, m) [0, m) [0, m) [0, m) = {( 1, 2,, 6 ) i [0, m), 1 i 6} be the 6 dimensionl MOD rel plne.

152 Higher nd Mixed Multidimensionl MOD Plnes 151 For tke m = 10 x = (9.2, 0.63, 8.42, 1.8, 4.8, 0) M, we cn dd them under +, M is group. Exmple 4.15: Let N = {( 1, 2, 3 ) i [0, 7); 1 i 3, +} be MOD rel multidimensionl structure. Let x = (6.03, 0.78, 1.05) nd y = (3.1, 4.1, 3.1) N. x + y = (2.13, 4.88, 4.15) nd x y = (4.693, 3.198, 3.255) N. This is the wy sum nd is performed on N. Exmple 4.16: Let T = {(b 1, b 2,, b 8 ) b i [0, 12); 1 i 8, +} be MOD rel multidimensionl plne under + is group. Let x = (10.3, 7.5, 0.71, 2.05, 1.8, 0.21, 6.32, 4.52) T x + x = (8.6, 3, 14.2, 4.1, 3.6, 0.42, 0.64, 9.04) T. x x = (0, 0, 0, 0, 0, 0, 0, 0) T. x x = (10.09, 8.25, , , 3.24, , , ) T.

153 152 Multidimensionl MOD Plnes However under product T is closed. Exmple 4.17: Let P = [0, 10) [0, 9) [0, 24) [0, 13) [0, 7) = {(x 1, x 2, x 3, x 4, x 5 ) x 1 [0, 10), x 2 [0, 9), x 3 [0, 24), x 4 [0, 13) nd x 5 [0, 7), +} be MOD mixed rel multidimensionl spce. We use the term mixed rel tht is the MOD intervls re different but ll re rel nd intervls re different. This is not possible in cse of rel plne only in cse of MOD rel intervls the vriety is very different. Let x = (7.2, 4.5, 13.6, 12.1, 6.2) nd y = (8.5, 6.7, 21.7, 10.7, 4.3) P. x + y = (5.7, 2.2, 11.3, 9.8, 3.5) P. x y = (1.2, 3.15, 7.12, , 5.66) P. Exmple 4.18: Let M = {([0, 10) [0, 5) [0, 10) [0, 6) [0, 6) [0, 5) [0, 2) [0, 4))} be MOD rel mixed multidimensionl spce. The sum nd re defined on M. It is pertinent to keep on record tht only in cse of MOD intervls we cn (1) hve the concept of mixed multi rel dimensionl spce. This is not possible using rels. (2) we hve infinitely mny such MOD mixed rel multidimensionl spces.

154 Higher nd Mixed Multidimensionl MOD Plnes 153 However these spces re groups under +. Under they re only semigroups. Further using both the opertions + nd does not stisfy the distributive lw only they re MOD multi mixed dimensionl spces re only MOD multidimensionl pseudo rings. We will illustrte them by exmples. Exmple 4.19: Let V = {([0, 7) [0, 4) [0, 12) [0, 10) [0, 6) [0, 5) [0, 4)), +, } be the MOD rel mixed multidimensionl pseudo ring. Let x = (3.7, 0.32, 6.1, 9.5, 5.2, 4.3, 3.1), y = (0.8, 0.7, 4.1, 0.2, 1.432, 3.2, 1.5) nd z = (4.2, 3.2, 10.6, 9.3, 4.3, 4.1, 3.1) V. Consider (x y) z = (2.96, 0.224, 1.01, 1.9, , 3.76, 0.65) (4.2, 3.2, 10.6, 9.3, 4.3, 4.1, 3.1) = (5.432, , 7.67, , 0.416, 2.015) I Consider x (y z) = (3.7, 0.32, 6.1, 9.5, 5.2, 4.3, 3.1) (3.36, 2.24, 7.46, 1.86, , 3.12, 0.65) = (5.432, , 1.506, 7.67, , 0.416, 2.015) II

155 154 Multidimensionl MOD Plnes Clerly I nd II re equl. Hence certinly the opertion is ssocitive. However x (y + z) x y + x z for x, y, z V. x (5, 3.9, 2.7, 9.5, 5.723, 2.3, 0.6) x y + x z = (4.5, 1.248, 4.47, 0.25, , , 1.61) I = (3.7, 0.32, 6.1, 9.5, 5.2, 4.3, 3.1) (0.8, 0.7, 4.1, 0.2, 1.423, 3.2, 1.5) + (3.7, 0.32, 6.1, 9.5, 5.2, 4.3, 3.1) (4.2, 3.2, 10.6, 9.3, 4.3, 4.1, 3.1) = (2.96, 0.224, 1.01, 1.9, , 3.76, 0.65) + (1.54, 1.024, 4.66, 8.35, 4.36, 2.63, 1.61) = (4.50, 1.248, 5.67, 0.25, , 1.39, 2.26) II I nd II re distinct so the ring is only pseudo ring. So only the MOD mixed rel multidimensionl spce is pseudo ring. Exmple 4.20: Let W = {([0, 14) [0, 10) [0, 14) [0, 5) [0, 80) [0, 5) [0, 10) [0, 23)), +, } be MOD rel mixed multidimensionl spce pseudo ring. Next we proceed on to define MOD multidimensionl complex spce. DEFINITION 4.2: Let M = (C n (m) C n (m) C n (m)) - 2 t-times; i F = m 1} be the MOD multidimensionl complex modulo integer spce. We will give exmples of them.

156 Higher nd Mixed Multidimensionl MOD Plnes 155 Exmple 4.21: Let V = (C n (10) C n (10) C n (10) C n (10)) be the MOD multidimensionl complex modulo integer spce. We see x = (9 + 2i F, 3 + 4i F, 6i F, 7) V nd y = ( i F, i F, i F, ) V. Clerly V under + is MOD multidimensionl complex modulo integer group nd (V, ) is MOD complex multidimensionl semigroup. Exmple 4.22: Let A = {(C n (7) C n (7) C n (7) C n (7) C n (7)), +} be the MOD multidimensionl complex modulo integer group. For if x = (2 + 3i F, 0.334i F, 2.701, i F, 6 + 2i F ) nd y = ( i F, 4 + 3i F, 5.11i F + 2, i F, 1 + 5i F ) A, then x + y = ( i F, i F, 5.11i F , 6i F, 0) A. This is the wy the opertion + is performed on A. Exmple 4.23: Let B = {(C n (10) C n (10) C n (10) C n (10) C n (10) C n (10)), } be the MOD multidimensionl complex modulo integer semigroup. Let x = ( i F, 4 + 2i F, i F, 4.25i F, , 0) nd

157 156 Multidimensionl MOD Plnes y = (0, 2 + 7i F, i F, 6.115i F, i F, i F ) B. x y = (0, 4 + 2i F, i F, i F, , 0) B. This is the wy product opertion is performed on the MOD multidimensionl complex modulo integer semigroup. Exmple 4.24: Let M = {(C n (5) C n (5) C n (5)), } be the MOD multidimensionl complex modulo integer semigroup. Next we proceed on to define MOD mixed multidimensionl complex modulo integer spce. DEFINITION 4.3: B = {(C n (m 1 ) C n (m 2 ) C n (m 3 ) C n (m s )) 2 tlest some m i s re distinct i F = (m i 1) for i = 1, 2,, s} is defined s the MOD mixed multidimensionl complex modulo integer spce. We will give some exmples of them. Exmple 4.25: Let T = {(C n (7) C n (10) C n (12) C n (7) C n (5))} be the MOD mixed multidimensionl complex modulo integer spce. Exmple 4.26: Let W = {(C n (11) C n (14) C n (8) C n (11) C n (14) C n (9))} be MOD mixed multidimensionl complex modulo integer spce.

158 Higher nd Mixed Multidimensionl MOD Plnes 157 We cn on these spces define + nd opertion. Under + opertion the MOD mixed multidimensionl complex modulo integer group nd under the MOD mixed complex modulo integer multidimensionl spce is semigroup. We will illustrte this sitution by some exmples. Exmple 4.27: Let M = {(C n (10) C n (7) C n (5) C n (4) C n (2)), +} be the MOD mixed multidimensionl complex modulo integer group. Let x = (3 + 7i F, 4i F + 3, i F, 3i F, 1 + i F ) nd y = (2 + 3i F, 4, 2i F, 0.2, i F ) M. x + y = (5, 4i F, i F, 3i F + 0.2, 1). This is the wy + opertion is performed on M. Exmple 4.28: Let T = {(C n (5) C n (7) C n (6) C n (11)), +} be the MOD mixed multidimensionl modulo integer group. T hs subgroups of finite order. T lso hs subgroups of infinite order. Exmple 4.29: Let W = {(C n (4) C n (6) C n (8) C n (10)), } be the MOD mixed multidimensionl complex modulo integer semigroup.

159 158 Multidimensionl MOD Plnes Let x = ( i F, i F, 6 + 2i F, i F ) nd y = ( i F, i F, i F, 5 + 2i F ) W. x y = ( i F, i F, i F, i F ) W. This is the wy product opertion is performed on W. Exmple 4.30: Let V = {(C n (17) C n (12) C n (15) C n (40)), } be the MOD mixed multidimensionl complex modulo integer semigroup. Infct we hve infinitely mny MOD mixed multidimensionl complex number spces. However we cnnot get ny mixed multidimensionl complex spces we hve only multidimensionl complex spces using C. Next we cn define MOD mixed multidimensionl complex modulo integer pseudo rings. DEFINITION 4.4: Let R = {(C n (m 1 ) C n (m 2 ) C n (m t )) i = (m i 1); 1 i t, +, } be the MOD mixed multidimensionl complex modulo integer spce. R under the opertions + nd is the MOD mixed multidimensionl complex modulo integer pseudo ring. We see if m 1 = m 2 = = m t = m then R is just MOD multidimensionl complex modulo integer pseudo ring. We will give exmples of them. 2 F

160 Higher nd Mixed Multidimensionl MOD Plnes 159 Exmple 4.31: Let W = {(C n (7) C n (9) C n (11) C n (13)); +, } be the MOD mixed multidimensionl complex modulo integer pseudo ring. Let x = ( i F, i F, i F, i F ) nd y = ( i F, i F, i F, i F ) nd z = ( i F, 4 + 2i F, i F, 7 + 2i F ) W. x + y = ( i F, i F, i F, i F ) nd W x y = ( i F, i F, i F, i F ) Consider x (y + z) = x y + x z = ( i F, i F, i F, i F ) + ( i F, i F, 6 + 6i F, i F ) = (5 + 2i F, 8i F, i F, i F ) I x (y + z) = ( i F, i F, i F, i 1F ) (4, i F, i F, i F ) = (5 + 2i F, i F, i F, i F ) II Clerly I nd II re distinct hence the distributive lw in generl is not true. Thus W is MOD mixed multidimensionl complex modulo integer pseudo ring.

161 160 Multidimensionl MOD Plnes Exmple 4.32: Let V = {(C n (12) C n (45) C n (24) C n (20) C n (24) C n (12)), +, } be the MOD mixed multidimensionl complex modulo integer pseudo ring. Clerly the MOD multidimensionl complex modulo integer vector spce over C n (m); however the MOD mixed multidimensionl complex modulo integer group is not vector spce for they re mixed. Next we proceed on to define these concepts using MOD multidimensionl neutrosophic structures. DEFINITION 4.5: Let I I I I V = {( R n (m) R n (m) R n (m) R n (m))} be the MOD multidimensionl neutrosophic spce. We will first give exmples of them. Exmple 4.33: Let I I I I M = {( R (8) R (8) R (8) R (8) )} n n n n be the MOD multidimensionl neutrosophic spce. Exmple 4.34: Let I I I I I V = {( R (20) R (20) R (20) R (20) R (20))} n n n n n be the multidimensionl neutrosophic spce. We define lgebric structure on them.

162 Higher nd Mixed Multidimensionl MOD Plnes 161 DEFINITION 4.6: Let V = {( ( R I ( m) R I ( m) R I ( m)), + } n n n be the MOD multidimensionl neutrosophic group. We will exmples of them. Exmple 4.35: Let I I I I V = {( R (9) R (9) R (9) R (9) ), +} n n n n be the MOD multidimensionl neutrosophic group. For if x = (3 + 4I, I, I, I) nd y = ( I, I, I, I) V. x + y = (7.5I, I, I, I) V. This is the wy + opertion is performed on the MOD multidimensionl neutrosophic group. Exmple 4.36: Let I I I I I M = {(R (10) R (10) R (10) R (10) R (10), + } n n n n n be the MOD multidimensionl neutrosophic group. Next we proceed on to define the MOD multidimensionl neutrosophic semigroup. I I DEFINITION 4.7: Let W = { ( Rn ( m) Rn ( m)), MOD neutrosophic multidimensionl semigroup. } be the We give exmples of them.

163 162 Multidimensionl MOD Plnes Exmple 4.37: Let I I I I I M = { (R (12) R (12) R (12) R (12) R (12)), } n n n n n be the MOD multidimensionl neutrosophic semigroup. Let x = ( I, 7.5I, 8.3, I, 6I) nd y = (8, I, I, 7I, I) M. x y = (0, 3I, I, 9I, 9I) M. This is the wy product opertion will be performed. Exmple 4.38: Let M = { R (10) I n I R n (10) I R n (10) R (10) I n I n R (10) I n R (10) be the MOD multidimensionl neutrosophic semigroup. I R n (10), } Let x = (6 + 3I, 7 + 4I, 1.8I, 4.5, I, I, 8I) nd y = (3I, 0.4I, 6 + 5I, 7 + 2I, 5, 6I, I) M; x y = (8 + 9I, 4.11I, 9.8I, 9I + 1.5, I, 4.8I, 0) M. This is the wy product opertions re performed on MOD multidimensionl neutrosophic semigroup. Next we proceed on to define MOD multidimensionl neutrosophic pseudo ring. DEFINITION 4.8: Let V = { ( R I ( m) R I ( m) R I ( m)), +, }; n n n

164 Higher nd Mixed Multidimensionl MOD Plnes 163 V is defined s the MOD multidimensionl neutrosophic pseudo ring. V is clled s the pseudo ring s the distributive lw is not true. We will illustrte this sitution by some exmples. Exmple 4.39: Let I I I M = { (R (5) R (5) R (5)), +, } n n n be the MOD multidimensionl neutrosophic pseudo ring. Let x = ( I, 0.8I, 0.9), y = (0.2I, I, I) nd z = (3, I, I) M. We find x + y, x y, xy + xz nd x(y + z) in the following. x + y = ( I, I, I) M. x y = (0.86I, 2.64I, I) M. Consider xy + xz = (0.86I, 2.64, I) + ( I, 0.36I, I) = ( I, 3I, I) I x(y + z) = ( I, 0.8I, 0.9) ( I, 0, 0) = ( I, 0, 0) II Clerly I nd II re distinct so the MOD multidimensionl neutrosophic pseudo ring does not stisfy the distributive lw.

165 164 Multidimensionl MOD Plnes Exmple 4.40: Let I I I I I S = { ( R (18) R (18) R (18) R (18) R (18)), +, } n n n n n be the MOD multidimensionl neutrosophic pseudo ring. Next we proceed on to define the notion of MOD mixed multidimensionl neutrosophic structures nd illustrte them by exmples. DEFINITION 4.9: Let S = { ( R I ( 1) I ( 2) I n m Rn m Rn ( mt )); tlest some of the m i s re distinct I 2 = I}; S is defined s the MOD mixed multidimensionl neutrosophic spce. We will give one or two exmples of them. Exmples 4.41: Let I I I I M = { (R (12) R (10) R (16) R (4)) } n n n n be the MOD mixed multidimensionl neutrosophic spce. Exmple 4.42: Let T = {( R (7) I n I R n (97) I R n (14) I R n (14) I R n (7) be the MOD mixed multidimensionl neutrosophic spce. I R n (14))} Now we define opertions on MOD mixed multidimensionl neutrosophic spces. DEFINITION 4.10: Let

166 Higher nd Mixed Multidimensionl MOD Plnes 165 P = { ( R I ( 1) I ( 2) I n m Rn m Rn ( mt )), +}. P is defined s the MOD mixed multidimensionl neutrosophic group. We will give exmples of them. Exmple 4.43: Let I I I I n n n n P = { (R (10) R (8) R (15) R (20)), +} be the MOD mixed multidimensionl neutrosophic group. Let x = ( I, I, I, 16.8I) nd y = ( I, I, I, I) P. x + y = (5 + 3I, I, I, I) P. This is the wy + opertion is performed on P. Exmple 4.44: Let I I I I I n n n n n T = { (R (12) R (6) R (12) R (9) R (11)), +} be the MOD mixed multidimensionl neutrosophic group. Next we proceed on to define MOD mixed multidimensionl neutrosophic group. DEFINITION 4.11: Let W = { ( R I ( 1) I ( 2) I n m Rn m Rn ( mr )); }; W under the product is defined s the MOD mixed multidimensionl neutrosophic semigroup.

167 166 Multidimensionl MOD Plnes Clerly W is of finite order nd W hs infinite zero divisors. Infct ll idels in W re of infinite order. W hs subsemigroups of finite nd infinite order which re not idels of W. We will illustrte this sitution by some exmples. Exmple 4.45: Let I I I I S = { (R (10) R (15) R (7) R (6)), } n n n n be the MOD neutrosophic mixed multidimensionl semigroup. Let x = ( I, I, 4 + 3I, I) nd y = (5I, 10I, 0.5I, 5I) S. x y = (7.5I, 3I, 3.5I, 2I) S. This is the wy the product opertion is performed on S. I n P 1 = {( R (10) {0} {0} {0}), } is subsemigroup of infinite order. Infct n idel of S. Let P 2 = {({0} Z 15 I {0} {0}); } S is subsemigroup of finite order nd is not n idel of S. P 3 = {( Z 10 I {0} {0} R I n (6) ), } is subsemigroup of S of infinite order. Clerly P 3 is not n idel of S. Thus S hs subsemigroups of infinite order which re not idel of S. Next we proceed on to give one more exmple. Exmple 4.46: Let

168 Higher nd Mixed Multidimensionl MOD Plnes 167 M = {( R (12) I n I R n (12) R (19) R (10) I n I n R (5) I n I R n (2)), } be the MOD mixed multidimensionl neutrosophic semigroup of infinite order. I n P 1 = {({0} {0} R (19) {0} {0} R (2) ), } M is the MOD mixed multidimensionl neutrosophic subsemigroup which is lso n idel of M. Let P 2 = {({0} {0} Z 19 {0} {0} {0}), } M is finite subsemigroup of M which is not n idel of M. I n Let P 3 = {( R (12) {0} {0} {0} Z 5 I {0}), } M be subsemigroup of M of infinite order. Clerly P 3 is not n idel of M only subsemigroup. Next we proceed on to describe nd develop the notion of MOD mixed multidimensionl neutrosophic pseudo rings. I I I DEFINITION 4.10: Let M = { ( Rn ( m1 ) Rn ( m2 ) Rn ( ms )); m i s re distinct 1 i s, +, }. M be defined s the MOD neutrosophic mixed multidimensionl pseudo ring. Clerly M is n infinite commuttive pseudo ring. We will give exmples of them. Exmple 4.47: Let I I I I M = { (R (10) R (7) R (4) R (10)); +, } n n n n be the MOD mixed multidimensionl neutrosophic pseudo rings. Clerly M hs infinite number of zero divisors, finite number of units nd idempotents. I n

169 168 Multidimensionl MOD Plnes M hs subrings of finite order s well s pseudo subrings of infinite order which re not idels. However ll idels of M re of infinite order. To this effect we will give pseudo subrings of infinite order which re not pseudo idels. I Consider P 1 = {( R n (10) {0} {0} {0})} M is pseudo subring of infinite order which is lso pseudo idel. I n Consider P 2 = {( R (10) {0} {0} Z 10 ), +, } M is pseudo subring of infinite order which is not n idel of M. I P 3 = {( Z 10 I R n (7) Z 4 {0}); +, } M is pseudo subring of infinite order which is not pseudo idel of M. Clerly M hs tlest 4 C C C 3 number of idels; however M hs severl pseudo subrings of infinite order which re not pseudo idels. M lso hs subrings of finite order. For T 1 = {(Z 10 {0} {0} {0}), +, } M, T 2 = {( Z 10 I {0} {0} {0}), +, } M, T 3 = {(Z 10 { Z 7 I } {0} {0}), +, } M nd so on re some of the subrings of finite order which re not pseudo. Exmple 4.48: Let I I I I n n n n N = { (R (5) R (2) R (4) R (6)), +, } be the MOD mixed multidimensionl neutrosophic pseudo ring.

170 Higher nd Mixed Multidimensionl MOD Plnes 169 N hs (1,1,1,1) to be unit. N hs infinite number of zero divisors. N hs severl idempotents. P 1 = {( R I n (5) {0} {0} idel. I R n (6), +, )} N is pseudo P 2 = {(Z 5 ( Z 2 I ) {0} {0}); +, } N is subring of finite order not pseudo. I P 3 = {(Z 5 R n (2) {0} {0}); +, } N is pseudo subring of infinite order which is not n idel. However ll MOD multidimensionl neutrosophic groups. I I I Under + sy V = { (R (m) R (m) R (m)); +} re n n n MOD neutrosophic vector spces over Z m if m is prime or MOD S-pseudo neutrosophic vector spces over Z m I or just MOD S-pseudo neutrosophic vector spces over R (m). Further it is pertinent to keep on record tht MOD mixed multidimensionl neutrosophic groups under + in generl re not MOD vector spces nd cnnot lso be mde into MOD neutrosophic vector spces s one common bse field or S-ring cnnot be determined. Next we proceed onto study MOD multidimensionl dul number spces nd MOD mixed multidimensionl dul number spces. DEFINITION 4.12: Let W = {(R n (m)g R n (m)g R n (m)g); g 2 = 0, t-times}. W is defined s the MOD multidimensionl dul number spce. W = {( 1 + b 1 g, 2 + b 2 g,, t + b t g); g 2 = 0, i b i [0, m); 1 i t}. W is finite collection. I n

171 170 Multidimensionl MOD Plnes First we will give exmples of them. Exmple 4.49: M = {(R n (10)g R n (10)g R n (10)g R n (10)g); g 2 = 0} = {( 1 + b 1 g, 2 + b 2 g, 3 + b 3 g, 4 + b 4 g) i b i [0, 10) g 2 = 0, 1 i 4} be the MOD multidimensionl dul number spce. M. x = ( g, g, g, g) Exmple 4.50: Let B = {(R n (19)g R n (19)g R n (19)g R n (19)g R n (19)g R n (19)g); g 2 = 0} = {( 1 + b 1 g, 2 + b 2 g, 3 + b 3 g, 4 + b 4 g, 5 + b 5 g, 6 + b 6 g) g 2 = 0, i, b i [0, 19); 1 i 6} We cn define lgebric opertions + nd on these MOD multidimensionl dul number spces. DEFINITION 4.13: Let P = {(R n (m)g, R n (m)g,, R n (m)g) g 2 = 0; 2 m < ; +}; P is group under ddition. P is defined s the MOD multidimensionl dul number group. P is n belin group of infinite order. We will give exmples of them. Exmple 4.51: Let M = {(R n (11)g R n (11)g R n (11)g R n (11)g) g 2 = 0; +} be the MOD multidimensionl dul number group M is MOD dul number vector spce over Z 11. However M is only Smrndche MOD dul number vector spce over ( Z 11 g ) or R n (11) or R n (11)(g).

172 Higher nd Mixed Multidimensionl MOD Plnes 171 Study of properties of enjoyed by these MOD vector spces is mtter of routine nd hence left s n exercise to the reder. Exmple 4.52: Let M = {(R n (12)g R n (12)g R n (12)g R n (12)g R n (12)g) g 2 = 0, +} be S-pseudo MOD multidimensionl dul number vector spce re the S-ring Z 12 or Z 12 g or [0, 12). Study of these properties re lso interesting nd work in this direction is left s n exercise for the reder. Exmple 4.53: Let T = {(R n (9)g R n (9)g R n (9)g R n (9)g); g 2 = 0, } be the MOD multidimensionl dul number semigroup. We see T hs subsemigroups which re zero squre subsemigroups. Let x = (4 + 3g, 2.5g, g, 2 + 4g) nd y = ( g, 4 + 2g, g + 1, g) T. x y = {( g, g, g, g)} T Let = (5g, 7.881g, g, 1.6g) nd b = (7g, 6.534g, 0.785g, 0.093g) T b = (0, 0, 0, 0). Thus if B = {( 1 g, 2 g, 3 g, 4 g) i [0, 9); 1 i 4, g 2 = 0} T is such tht B is subsemigroup nd infct B is n idel.

173 172 Multidimensionl MOD Plnes B is infct zero squre semigroup. Exmple 4.54: Let S = {(R n (12)g R n (12)g R n (12)g R n (12)g R n (12)g), g 2 = 0, } be the MOD multidimensionl dul number semigroup. S hs MOD idels which re of infinite order. Infct S hs subsemigroups of finite order none of which re idels. Infct S hs subsemigroups of infinite order which re lso not idels only subsemigroups. Let W = {(R n (12)g {0} {0} Z 12 Z 12 g ); g 2 = 0} be only subsemigroup of S nd is not idel of S. Now we just show V = {(Z 12 {0, 6} {0, 3, 6, 9} {0, 4g, 8g} Z 12 g ) g 2 = 0, } S is only subsemigroup of finite order. Next we proceed on to define the notion of MOD multidimensionl dul number pseudo ring. DEFINITION 4.14: Let R = {(R n (m)g R n (m)g R n (m)g) g 2 = 0; 2 m <, +, } be defined s the MOD multidimensionl dul number pseudo ring. Clerly R is of infinite order commuttive but does not stisfy the distributive lws. All pseudo idels in R re of infinite order. However R hs finite subrings which re not pseudo. Infct R hs infinite number of zero divisors.

174 Higher nd Mixed Multidimensionl MOD Plnes 173 We will illustrte this sitution by some exmples. Exmple 4.55: Let P = {(R n (10)g R n (10)g R n (10g) R n (10)g) g 2 = 0, +, } be the MOD multidimensionl dul number pseudo ring. Let T 1 = {(Z 10 Z 10 Z 10 {0}), +, } P be the MOD multidimensionl dul number subring of finite order. T 2 = {(R n (10)g {0} {0} {0} {0}); +, } is MOD multidimensionl dul number pseudo subring which is pseudo idel of P. Let W 1 = {({0} R n (10)g {0} {0} Z 10 ); g 2 = 0, +, } P be the MOD multidimensionl dul number pseudo subring of infinite order which is not pseudo idel of P. Exmple 4.56: Let M = {(R n (17)g R n (17)g R n (17)g R n (17)g); g 2 = 0, +, } be the MOD multidimensionl dul number pseudo ring. P 1 = {(R n (17)g {0} {0} {0}), +, } is the MOD multidimensionl pseudo idel. P 2 = {(R n (7) {0} {0} {0}) +, } is only MOD multidimensionl pseudo subring of infinite order but is not pseudo idel of M. P 3 = {(Z 7 Z 7 g {0} {0}), +, } is MOD multidimensionl subring of finite order which is not pseudo ring. Thus these MOD multidimensionl dul number pseudo rings hve subrings which re not pseudo.

175 174 Multidimensionl MOD Plnes We see next it is mtter of routine to build MOD mixed multidimensionl dul number groups, semigroups nd pseudo rings using MOD mixed multidimensionl dul number spces. This will be illustrted by n exmple or two. Exmple 4.57: Let M = {(R n (5)g R n (10)g R n (12)g R n (9)g); g 2 = 0, +} be the MOD mixed multidimensionl dul number spce group. Clerly M is of infinite order but M hs subgroups of both finite nd infinite order. T 1 = {(R n (15)g {0} {0} {0}) g 2 = 0, +} M is subgroup of M of infinite order. T 2 = {(Z 5 Z 10 Z 12 Z 9 ); +, } M is subgroup of finite order. Exmple 4.58: Let B = {(R n (11)g R n (12)g R n (10)g R n (11)g R n (24)g R n (12)g) g 2 = 0, +} be the MOD mixed multidimensionl dul number spce group. B hs both subgroups of finite nd infinite order. Exmple 4.59: Let M = {(R n (10)g R n (6)g R n (2)g R n (15)g) g 2 = 0, } be the MOD mixed multidimensionl dul number semigroup. M is of infinite order nd is commuttive. All idels of M re of infinite order but hs subsemigroups of both finite nd infinite order.

176 Higher nd Mixed Multidimensionl MOD Plnes 175 Infct M hs infinite order subsemigroups which re not idels. M hs infinite number of zero divisors. N 1 = {(Z 10 Z 6 {0} {0}), } M is subsemigroup of finite order. N 2 = {(R 6 (10)g {0} {0} {0}) g 2 = 0, } M is n infinite subsemigroup which is n idel of M. N 3 = {(R n (10) Z 6 g {0} {0}); g 2 = 0, } is the subsemigroup of infinite order but is not idel of M. Exmple 4.60: Let M = {(R n (18)g R n (40)g R n (12)g R n (7)g R n (492)g) g 2 = 0, } be the MOD mixed multidimensionl dul number semigroup. N 1 = {(Z 8 {0} {0} {0} Z 492 ) } be the MOD subsemigroup of finite order which is not n idel of M. Next we proceed on to give exmple of MOD mixed multidimensionl dul number pseudo ring. Exmple 4.61: Let M = {(R n (16)g R n (7)g R n (22)g R n (45)g R n (11)g R n (7)g) g 2 = 0, +, } be the MOD mixed multidimensionl dul number spce pseudo ring. M hs subrings of finite order which re not pseudo. All subrings of infinite order re pseudo. However ll pseudo idels

177 176 Multidimensionl MOD Plnes re of infinite order. But M hs pseudo subrings of infinite order re not idels. P 1 = {(Z 16 {0} {0} {0} R n (11) Z 7 ), +, } M is pseudo subring of infinite order but is not n idel of P 1. P 2 = {({0} R n (7)g {0} {0} {0}), g 2 = 0, +, } is n pseudo idel of M. P 3 = {({0} {0} R n (22) {0} {0} {0}), +, } is pseudo subring of M of infinite order but is not n idel of M. Exmple 4.62: Let S = {(R n (11)g R n (18)g R n (17)g); g 2 = 0, +, } be the MOD mixed multidimensionl dul number spce pseudo ring. S hs infinite order pseudo subring which re not pseudo idels. S hs subrings of finite order which is not pseudo. Next we proceed on to give exmples of MOD multidimensionl specil dul like number spce groups, semigroups nd pseudo rings. Exmple 4.63: Let S = {(R n (10)h R n (12)h R n (40)h R n (12)h); h 2 = h, +} be the MOD mixed multidimensionl specil dul like number spce group. S hs both finite nd infinite order subgroups. V 1 = {(Z 10 Z 12 h {0} {0}) h 2 = h, +} is finite order subgroup. V 2 = {(R n (10)h {0} {0} {0}), h 2 = h, +} is n infinite order subgroup.

178 Higher nd Mixed Multidimensionl MOD Plnes 177 Exmple 4.64: Let M = {(R n (16)h R n (18)h R n (19)h R n (47)h R n (27)h); h 2 = h, +} be the MOD mixed multidimensionl specil dul like number group of infinite order. M hs subgroups of both of finite nd infinite order. Exmple 4.65: Let T = {(R n (10)h R n (10)h R n (10)h); h 2 = h, +} be the MOD multidimensionl specil dul like number spce group. T hs both subgroups of finite nd infinite order. Exmple 4.66: Let N = {(R n (20)h R n (20)h R n (20)h R n (20)h R n (20)h); h 2 = h, +} be the MOD multidimensionl specil dul like number spce group. N hs both subgroups of finite order s well s subgroups of infinite order. Exmple 4.67: Let P = {(R n (11)h R n (11)h R n (11)h R n (11)h); h 2 = h, } be the MOD multidimensionl specil dul like number semigroup. P hs subsemigroups of finite order. B 1 = {(Z 11 Z 11 {0} {0}), } is subsemigroup of finite order which is not n idel.

179 178 Multidimensionl MOD Plnes B 2 = {(R n (11)h {0} {0} {0}) } is subsemigroup of infinite order which is n idel. B 3 = {(R n (11) R n (11) {0} {0}), } is subsemigroup of infinite order which is not n idel. Exmple 4.68: Let M = {(R n (10)h R n (18)h R n (128)h); h 2 = h, } be the MOD mixed multidimensionl specil qusi dul like spce semigroup. N 1 = {(R n (10) {0} {0}) } is the infinite order subsemigroup. N 2 = {(Z 10 {0} Z 128 ), } is finite order subsemigroup. N 3 = {({0} {0} R n (128)h), h 2 = h, } is n infinite order subsemigroup which is n idel of M. Exmple 4.69: Let S = {(R n (20)h R n (45)h R n (20)h R n (2)h R n (4)h); h 2 = h, } be the MOD mixed multidimensionl specil dul like number semigroup. S hs idels ll of which re of infinite order. S hs subsemigroups of finite order which re not idels. S hs subsemigroups of infinite order which re not idels. Exmple 4.70: Let S = {(R n (10)h R n (10)h R n (10)h R n (10)h); h 2 = h, +, } be the MOD multidimensionl specil qusi dul like number spce pseudo ring.

180 Higher nd Mixed Multidimensionl MOD Plnes 179 S hs subrings of finite order which re not pseudo. S hs subrings of infinite order which re pseudo. Some pseudo subrings of infinite order re not idels. P 1 = {(Z 10 {0} {0} {0}), +, } is subring of finite order which re not pseudo. P 2 = {(R n (10) {0} {0} Z 10 ), +, } is pseudo subring of infinite order but is not n idel of S. However P 3 = {({0} R n (10)h R n (10)h {0}); h 2 = h, +, } is multidimensionl specil qusi dul number spce pseudo subring which is lso of infinite order nd is pseudo idel. Exmple 4.71: Let M = {(R n (8)h R n (10)h R n (13)h R n (16)h); h 2 = h, +, } be MOD mixed multidimensionl specil dul number spce pseudo ring. M hs both infinite nd finite order subrings which re not idels. T 1 = {(Z 8 {0} {0} Z 16 ); +, } is subring of finite order which is not pseudo. T 2 = {(R n (8) R n (10) {0} {0}), +, } is subring of infinite order which is pseudo but is not n idel. T 3 = {(R n (8)h R n (10)h {0} {0}); h 2 = h, +, } is pseudo subring of infinite order which is pseudo idel. Now we just give few exmples of MOD multidimensionl specil qusi dul number spce groups, semigroups nd pseudo rings.

181 180 Multidimensionl MOD Plnes We will lso give exmples of MOD mixed multidimensionl specil qusi dul number spce groups, semigroups nd pseudo rings. Exmple 4.72: Let M = {(R n (12)k R n (12)k R n (12)k R n (12)k); k 2 = 11k} be the MOD multidimensionl specil qusi dul number spce. Exmple 4.73: Let N = {(R n (10)k 1 R n (15)k 2 R n (27)k 3 R n (9)k 4 ), k = 9k 1, k = 14k 2, k = 26k 3, k = 8k 4 } be the MOD mixed multidimensionl specil qusi dul number spce. Exmple 4.74: Let P = {(R n (8)k R n (8)k R n (8)k R n (8)k); k 2 = 7k, +} be the MOD multidimensionl specil qusi dul number spce group. P is of infinite order. P hs subgroups of both finite nd infinite order. Exmple 4.75: Let M = {(R n (9)k 1 R n (20)k 2 ) k 1 = 8k 1 nd k 2 = 19k 2, +} be the MOD mixed multidimensionl specil qusi dul number spce group. P 1 = {Z 9 Z 20 } is subgroup of finite order. P 2 = {R n (9)k 1 {0}} is subgroup of infinite order.

182 Higher nd Mixed Multidimensionl MOD Plnes 181 M hs severl subgroups of finite order s well s subgroups of infinite order. Exmple 4.76: Let S = {(R n (10)k R n (10)k R n (10)k), k 2 = 9k, } be the MOD multidimensionl specil qusi dul number spce semigroup. All idels of S re of infinite order. However S hs subsemigroups of infinite order which re not idel. Finlly ll subsemigroups of finite order re not idels. P 1 = {(R n (10)k {0} {0})} is n idel of S. P 2 = {(R n (10) R n (10) {0})} is subsemigroup of infinite order nd is not n idel of S. P 3 = {(Z 10 Z 10 Z 10 ) } is subsemigroup of finite order. Exmple 4.77: Let B = {(R n (12)k 1 R n (13)k 2 R n (15)k 3 R n (24)k 4 ) 2 k 2 = 12k 2, 2 k 3 = 14k 3, 2 k 1 = 11k 1, k = 23k 4, } be the MOD mixed multidimensionl specil qusi dul number spce semigroup. B 1 = {Z 12 Z 13 Z 15 Z 24, } is subsemigroup of finite order. B 2 = {R n (12) R n (13) R n (15) R n (24), } is MOD subsemigroup of infinite order which is not n idel. 2 4

183 182 Multidimensionl MOD Plnes B 3 = {(R n (12)k 1 {0} {0} {0}), } is MOD subsemigroup of infinite order which is n idel of B. Exmple 4.78: Let W = {(R n (13)k R n (13)k R n (13)k R n (13)k) k 2 = 12k, +, } be the MOD multidimensionl specil qusi dul number spce pseudo ring. P 1 = {Z 13 Z 13 {0} {0}, +, } is finite subring which is not pseudo. P 2 = {(R n (13)k {0} {0} {0}); k 2 = 12k, +, } is pseudo subring of infinite order which is lso pseudo idel. P 3 = {(R n (13) R n (13) {0} {0}) +, } is pseudo subring of infinite order which is not n idel of W. Exmple 4.79: Let M = {(R n (12)k 1 R n (12)k 2 R n (12)k 3 ), 2 k 2 = 26k 2 nd k 1 = 11k 1, k = 48k 3, +, } be the MOD mixed multidimensionl specil qusi dul number pseudo ring. V 1 = {R 12 R 27 R 49 ; +, } is subring of finite order which is not pseudo. V 2 = {(R n (12) R n (27) {0}), +, } is subring of infinite order which is pseudo but is not n idel. 2 V 3 = {(R 3 (12)k 1 {0} {0}), k 1 = 11k 1, +, } is pseudo subring of infinite order which is lso pseudo idel of M.

184 Higher nd Mixed Multidimensionl MOD Plnes 183 Now we proceed on to develop the notion of MOD mixed multidimensionl spces nd groups, semigroups nd pseudo rings built using them. Exmple 4.80: Let B = {(R n (10) I R n (12) R n (15) R n (7)g); g 2 = 0, I 2 = I} be the MOD mixed multidimensionl spce. Exmple 4.81: Let B = {(R n (8)k R n (9)h I n R (14) R n (10)g C n (20) R n (17)) g 2 = 0, k 2 = 7k, h 2 = h, I 2 = I, be the MOD mixed multidimensionl spce. Exmple 4.82: Let M = {(R n (7) I R n (20) R n (5)), +} 2 i F = 19} be the MOD mixed multidimensionl spce group. M hs subgroups of both finite nd infinite order. P 1 = {(Z 7 {0} Z 5 ), +} is subgroup of finite order. P 2 = {(R n (7) {0} R n (5)), +} is subgroup of infinite order. Exmple 4.83: Let M = {(R n (7) I R n (20) R n (15)g R n (17)h C n (20) R n (7)); +} be MOD mixed multidimensionl group. Exmple 4.84: Let

185 184 Multidimensionl MOD Plnes V = {(R n (5)h R n (20)g I R n (40)), h 2 = h, g 2 = 0, I 2 = I, } be the MOD mixed multidimensionl semigroup. Clerly V is of infinite order. All idels of V re of infinite order. There re subsemigroups of finite order s well s infinite order which re not idels. W 1 = {(Z 5 Z 20 Z 40 ); } is subsemigroup of finite order. Clerly W 1 is not n idel of V. W 2 = {(R n (5) R n (20) {0}), } is gin subsemigroup of infinite order which is not n idel of V. I W 3 = {({0} {0} R n (40)), } is subsemigroup of infinite order which is lso n idel of V. Exmple 4.85: Let M = {(R n (5) R n (10) R (5) C n (10) R n (10)g I n R n (10)h) h 2 = h, g 2 = 0, I 2 = 0, 2 i F = 9, } be the MOD mixed multidimensionl spce semigroup. M hs infinite number of zero divisors, some finite number of units nd idempotents. However M hs subsemigroups of both finite nd infinite order which re not idels. But ll idels of M re of infinite order. Exmple 4.86: Let B = {( R (7) R n (10) R n (5)g R n (5)h R n (10)k I n C n (10)) I 2 = I, g 2 = 0, h 2 = h, k 2 = 9k, be the MOD mixed multidimensionl pseudo ring. 2 i F = 9, +, }

186 Higher nd Mixed Multidimensionl MOD Plnes 185 B is of infinite order. All idels of B re of infinite order nd re pseudo. B hs subrings of finite order which re not idels. B hs subrings of infinite order which re not pseudo idels but only pseudo subrings. Let V 1 = {(Z 7 {0} {0} {0} {0} C(Z 10 )), +, } is subring of B of finite order. Clerly V 1 is not n idel of B. V 2 = {({0} Z 10 Z 5 {0} {0} {0}), +, } is gin subring of B nd not n idel of B. V 3 = {(R n (7) {0} {0} {0} {0} R n (10)), +, } is subring of B nd is infct pseudo subring of B but V 3 is not pseudo idel of B. Let V 4 = {({0} {0} R n (5) R n (5) R n (10) {0}), +, } is pseudo subring of B of infinite order but is not pseudo idel of B. I Let V 5 = {( R n (7) R n (10) {0} {0} {0} {0}), +, } is pseudo subring of B of infinite order which is lso pseudo idel of B. Let V 6 = {({0} {0} {0} {0} {0} C n (10)), +, } be the pseudo subring of infinite order which is lso pseudo idel of B. Exmple 4.87: Let T = {(R n (7)h R n (7)g R (7) C n (7) R n (7)k) h 2 = h, g 2 = 0, k 2 = k; I 2 = I, be the MOD mixed multidimensionl pseudo ring. I n 2 i F = 6, +, } Clerly T cn be relized s the MOD mixed multidimensionl pseudo liner lgebr over the field Z 7.

187 186 Multidimensionl MOD Plnes In view of ll these we mke the following theorem. THEOREM 4.5: Let V = {(R n (m)k R n (m) R n (m)g R n (m)h R n (m)k C n (m)) g 2 = 0, h 2 = h, k 2 2 = (m 1)k, i F = m 1, +, } be the MOD mixed multidimensionl pseudo ring. V is MOD mixed multidimensionl pseudo liner lgebr over Z m if nd only if m is prime V is MOD mixed multidimensionl S-pseudo liner lgebr if m is non prime nd Z m is S-ring. Proof is direct nd hence left s n exercise to the reder. Thus we cn hve MOD mixed multidimensionl pseudo rings. Exmple 4.88: Let M = {(R n (5) R n (7) R n (8)g R n (11)h R n (5)) g 2 = 0, h 2 = h, +, } be the MOD mixed multidimensionl pseudo ring. Let x = (0.7, 5, g, h, 3) nd y = (4, 2, 4g, 5h, 2) M x y = (2.8, 3, 0, 4h, 4) M Clerly M is not mixed multidimensionl vector spce or pseudo liner lgebr over ny common field or S ring. We cn work in this direction to get mny other interesting properties bout these newly constructed MOD structures.

188 Chpter Five SUGGESTED PROBLEMS In this chpter we suggest few problems for the reder. Some of these problems re t reserch level nd some re simple nd innovtive. 1. Cn we sy use of MOD rel intervls [0, m); 2 m < insted of (, ) in pproprite problems re dvntgeous? 2. Cn one clim use of MOD rel intervls in storge system will sve time nd economy? 3. Specify the dvntges of using MOD rel intervls [0, m) in the plce of (, ); 2 m <. 4. Cn we sy MOD rel intervls is cpble of better results? 5. Wht re the dvntges of mking the negtives into the MOD intervl [0, m)? 6. Show the MOD rel intervl trnsformtion is periodic one mpping periodiclly infinite number of points to single point in [0, m).

189 188 Multidimensionl MOD Plnes 7. Cn we sy [0, 1) is the MOD fuzzy intervl which cn depict (, )? 8. Wht is the mjor difference between the use of fuzzy intervl [0, 1] nd the MOD fuzzy intervl [0, 1)? 9. Show the MOD intervl [0, 9) cn be got s multidimensionl projection of (, ) to [0, 9). Cn we hve n inverse of I η : (, ) [0, 9)? 10. Show in problem 9 I η (9t + s) = s where s [0, 9) nd t (, ). 11. In problem 9 if t Z will mp I η : (, ) [0, 9) complete? Justify your clim. 12. Let I η : (, ) [0, 18) be the MOD rel trnsformtion. i. Wht re the specil fetures enjoyed by I η? ii. Wht is zeros of I η? Is it nontrivil? iii. When is I η (m) = 0? Show I η (m) =. iv. Cn I η hve T η such tht I η T η = T η I η? v. Cn T η be mp or pseudo specil type of mp? vi. Show point in [0, 18) is mpped by T η into infinitely mny points in (, ). vii. Enumerte ll the specil fetures enjoyed by T η. viii. Prove T η is not the clssicl mp or function but such study is lso needed. 13. Wht mkes T η : [0, m) (, ) essentil? 14. Prove mps like T η : [0, m) (, ) re lso well defined in their own wy.

190 Suggested Problems Obtin ny other specil properties of MOD specil pseudo functions, T η : [0, m) (, ). 16. Give the MOD mp from η z : Z n Z. Enumerte the properties enjoyed by η z. 17. Show S = {[0, m), } is MOD non ssocitive semigroup of infinite order. 18. Prove S = {[0, 15), } cn hve zero divisors. i. Cn S hve idempotents? ii. Does S contin S idempotents? iii. Cn S hve S zero divisors? iv. Cn S hve infinite number of zero divisors? v. Cn S hve S units? vi. Cn S hve infinite number of units? vii. Cn S hve idels of finite order? viii. Cn S hve subsemigroups of finite order? ix. Cn S hve S-idels? 19. Let S 1 = {[0, 23), } be the MOD rel intervl non ssocitive semigroup. Study questions i to ix of problem 18 for this S Let S 2 = {[0, 24), } be the rel MOD semigroup. Study questions i to ix of problem 18 for this S Let R = {[0, 7), } be the rel MOD semigroup. Prove R hs triple which is non ssocitive. 22. Let G = {[0, 9), +} be MOD rel intervl group.

191 190 Multidimensionl MOD Plnes i. Prove G is of infinite order. ii. Prove G hs subgroups of infinite order. iii. Find 3 subgroups of infinite order in G. iv. Find 5 subgroups of finite order. 23. Let G = {[0, 12), +} be MOD rel intervl group. Study i to iv of questions of problem 22 for this G. 24. Let R = {[0, 5), +, } be the MOD rel intervl pseudo ring. i. Prove R is non ssocitive. ii. Cn R hve zero divisors? iii. Cn R hve S zero divisors? iv. Find ll units in R. v. Cn R hve idempotents? vi. Cn R hve S units? vii. Cn R hve S idels? viii. Cn R hve S subrings of infinite order which re not idels? ix. Cn R hve idels of finite order? x. Find two idels of infinite order in S. 25. Let R 1 = {[0, 25), +, } be the MOD intervl pseudo ring. i. Find ll zero divisors in R 1. ii. Cn R 1 hve S-zero divisors? iii. Cn R 1 hve idempotents? iv. Cn R 1 hve S-idempotents? v. Cn R 1 hve units? vi. Is it possible for R 1 to hve S-units? vii. Find ll finite subrings of R 1. viii. Cn R 1 hve idels of finite order? ix. Find ll pseudo subrings of infinite order. 26. Let M = {[0, 17), +, } be the MOD rel pseudo ring. Study questions i to ix of problem 25 for this M.

192 Suggested Problems Enumerte ll the specil fetures enjoyed by MOD rel pseudo rings. 28. Prove using the MOD complex finite modulo integer [0, 24)i F one cnnot build MOD complex semigroups or MOD complex pseudo rings? 29. Enumerte the specil fetures enjoyed by the MOD complex modulo integer group. B = {[0, 12)i F, +}. 30. Develop ll the specil fetures enjoyed by S={[0, m)i F, +}. 31. Study ll the properties enjoyed by P = {[0, 13)i F, +}. 32. Let M = {[0, 24)I I 2 = I, +} be the MOD neutrosophic intervl group. i. Find ll subgroups of finite order in M. ii. Find ll subgroups of infinite order. iii. Find ny other specil feture enjoyed by M. 33. Let V = {[0, 47)I I 2 = I; +} be the MOD neutrosophic intervl group. Study questions i to iii of problem 32 for this V. 34. Let T = {[0, 24)I, I 2 = I, } be the MOD intervl neutrosophic semigroup. i. Cn T hve zero divisors? ii. Cn T hve idempotents? iii. Cn T hve S-zero divisors? iv. Cn T hve S-idempotents? v. Cn T hve units? vi. Is I the unit of T? vii. Cn T hve S-units? viii. Is T Smrndche semigroup?

193 192 Multidimensionl MOD Plnes ix. Prove T hs finite order subsemigroups. x. Prove T hs idels ll of them re only of infinite order. 35. Let S = {[0, 28)I; I 2 = I, } be the MOD neutrosophic intervl semigroup. i. Is S S-semigroup? ii. Cn S hve S-zero divisors? iii. Cn S hve zero divisors which re not S-zero divisors? iv. Cn S hve S-idels? v. Does S hve idels which re not S-idels? vi. Cn S hve S-units? vii. Cn S hve subsemigroups which of finite order which is not n idel? 36. Let P = {[0, 45)I; I 2 = I, } be the MOD neutrosophic intervl semigroup. Study questions i to vii of problem 35 for this P. 37. Let L = {[0, 492)I; I 2 = I, } be the MOD neutrosophic intervl semigroup. Study questions i to vii of problem 35 for this L. 38. Let R = {[0, 12)I, I 2 = I, +, } be the MOD neutrosophic intervl pseudo ring. i. Is R S-pseudo ring? ii. Cn R hve idels of finite order? iii. Cn R hve S-subrings of finite order? iv. Cn R hve S-idels? v. Cn R hve zero divisors? vi. Is R ring with unit? vii. Cn R hve S-units? viii. Cn R hve idempotents? ix. Cn R hve S-idempotents? x. Cn R hve S-zero divisors? xi. Cn R hve idels which re not S-idels?

194 Suggested Problems Let M = {[0, 17)I, I 2 = I, +, } be the MOD neutrosophic intervl pseudo ring. Study questions i to xi of problem 38 for this M. 40. Let B = {[0, 48)I; I 2 = I, +, } be the MOD neutrosophic intervl pseudo ring. Study questions i to xi of problem 38 for this B. 41. Let M = {[0, 9)g; g 2 = 0, +} be the MOD dul number intervl group. i. Prove M is of infinite order. ii. Cn M hve finite order subgroup? iii. Cn M hve subgroups of infinite order? 42. H = {[0, 13)g, g 2 = 0, +} be the MOD dul number intervl group. Study questions i to iii of problem 41 for this H. 43. Let S = {[0, 18)g, g 2 = 0, +} be the MOD dul number intervl group. Study questions i to iii of problem 41 for this S. 44. Let T = {[0, 19)g, g 2 = 0, } be the MOD dul number intervl semigroup. i. Find zero divisors if ny in T. ii. Cn T hve S-zero divisors? iii. Is T monoid? iv. Cn T hve idels of finite order? v. Cn T hve subsemigroups of finite order? vi. Cn T hve S-idempotents? vii. Cn T hve idempotents which re not S-idempotents?

195 194 Multidimensionl MOD Plnes viii. Is T S-semigroup? ix. Cn T hve S-idels? 45. Let S = {[0, 48)g g 2 = 0, } be the MOD dul semigroup. i. Study questions i to iii of problem 41 for this S. ii. Prove questions vi, vii nd viii of problem 43 re not true. 46. Let W = {[0, 24)g g 2 = 0, +, } be the MOD dul number ring. i. Cn W hve units? ii. Is W zero squre ring? iii. Find ll properties enjoyed by W. iv. Cn W hve finite order subrings? v. Cn W hve idels of finite order? vi. Obtin some specil fetures enjoyed by W. vii. Cn W hve subrings which re not idels? Justify. 47. Let S = {[0, 12)g g 2 = 0, +, } be the MOD dul number pseudo ring. Study questions i to vii of problem 46 for this S. 48. Let M = {[0, 49)g, g 2 = 0, +, } be the MOD dul number intervl ring. i. Cn M be S-pseudo ring? ii. Study questions i to vii of problem 46 for this S. 49. Let T = {[0, 45)h, h 2 = h, } be the MOD specil dul like number semigroup. i. Show T hs finite order subgroups. ii. Is T monoid? iii. Wht is the unit element of T? iv. Is h the unit element of T? v. Cn T hve idels of finite order?

196 Suggested Problems 195 vi. Is every infinite subsemigroup n idel? vii. Is T S-semigroup? viii. Cn T hve S-idempotents? ix. Cn T hve S-unit? x. Obtin ny other specil feture enjoyed by T. 50. Let M = {[0, 45)h, h 2 = h, +} be the MOD specil dul like number group. Obtin ll the specil fetures enjoyed by M. 51. If [0, 45)h in 50 is replced by [0, 23)h will these groups enjoy different properties. 52. Let V = {[0, 48)h, h 2 = h, } be the MOD specil dul like number semigroup. Study questions i to x of problem 49 for this V. 53. Let S = {[0, 47)h, h 2 = h, } be the MOD specil dul like number intervl semigroup. Study questions i to x of problem 49 for this S. 54. Let X = {[0, 144)h, h 2 = h, +, } be the MOD specil dul like number intervl pseudo ring. i. Is X S-ring? ii. Cn S hve S-idels? iii. Does S contin units? iv. Cn S hs idels of finite order? v. Cn S hve subrings of finite order? vi. Cn S hve S-units? vii. Cn S hve S-idempotents? viii. Cn S hve zero divisors which re not S-zero divisors? ix. Obtin some very specil fetures ssocited with S. 55. Let H = {[0, 43)h, h 2 = h, +, } be the MOD specil dul like number intervl pseudo ring.

197 196 Multidimensionl MOD Plnes Study questions i to ix of problem 54 for this H. 56. Let Y = {[0, 192)h, h 2 = h, +, } be the MOD specil dul like number intervl pseudo ring. Study questions i to ix of problem 54 for this Y. 57. Let Z = {[0, 12)k, k 2 = 11k, +, } be the MOD specil qusi dul number intervl pseudo ring. Study questions i to ix of problem 54 for this Z. 58. Let M = {[0, 25)k, k 2 = 24k, +, } be the MOD qusi dul number intervl pseudo ring. Study questions i to ix of problem 54 for this Z. 59. Let V = {[0, 48)k, k 2 = 47k, +, } be the MOD qusi dul number intervl pseudo ring. Study questions i to ix of problem 54 for this V. 60. Let Z = {[0, 28)k, k 2 = k, } be the MOD qusi dul number intervl semigroup. i. Cn Z hve S-zero divisors? ii. Is Z monoid? iii. Is Z S-semigroup? iv. Find idempotents if ny in Z. v. Cn Z hve S-idempotents? vi. Cn Z hve finite order subsemigroups? vii. Cn idels of Z be of finite order? viii. Find idels of Z. ix. Cn Z hve subsemigroup of infinite order which is not n idel of Z? x. Obtin ny other specil property enjoyed by Z.

198 Suggested Problems Let P = {[0, 17)k, k 2 = 16k, } be the MOD specil qusi dul number semigroup. Study questions i to x of problem 60 for this P. 62. Let S = {[0, 24)k, k 2 = 23k, } be the MOD specil qusi dul number intervl semigroup. Study questions i to x of problem 60 for this S. 63. Let V = {(x 1, x 2, x 3, x 4 ) x i [0, 15); 1 i 4, +} be the MOD rel mtrix group. i. Prove V hs subgroups of finite order. ii. Prove V hs subgroups of infinite order. iii. Give ny other interesting property ssocited with V. 64. Let W = i [0.97); 1 i 24, +} be the MOD rel number intervl group. Study questions i to iii of problem 63 for this W.

199 198 Multidimensionl MOD Plnes 65. Let M = i [0, 10) X n 1 i 10} be the MOD rel mtrix intervl semigroup. i. Prove M hs infinite number of zero divisors. ii. Cn M hve S-zero divisors? iii. Cn M hve S-units? iv. Cn M hve infinite number of idempotents? v. Does M contin S-idempotents? vi. Find subsemigroup of finite order in M. vii. Cn M hve idels of finite order? viii. Find ll idels of M. ix. Obtin ny other specil property enjoyed by M. x. Cn M be S-semigroup? 66. Let P = i [0, 23), 1 i 40, n } be the MOD rel intervl mtrix semigroup. Study questions i to x of problem 65 for this P.

200 Suggested Problems Let M = i [0, 28); 1 i 18, n } be the MOD rel mtrix intervl semigroup. Study questions i to x of problem 65 for this M. 68. Let B = {( 1, 2,, 10 ) i [0, 19); 1 i 10, +, } be the MOD rel mtrix intervl pseudo ring. i. Is B S-ring? ii. Cn B hve idels of finite order? iii. Cn B hve S-idels? iv. Cn B hve subrings of finite order? v. Does B contin zero divisors which re not S-zero divisors? vi. Does B contin S-idempotents? vii. Cn B hve S-units? viii. Find ny other specil feture enjoyed by B. ix. Prove B hs pseudo subrings of infinite order which re not idels. x. If P 1 = {( 1, 0 0) i [0, 19), +, } B. Find the quotient pseudo ring B/P Let X = { i [0, 44); 1 i 9, +, n } be the MOD rel intervl mtrix pseudo ring. i. Study questions i to x of problem 68.

201 200 Multidimensionl MOD Plnes ii. If the nturl product n is replced by show is non commuttive MOD pseudo ring. Study questions i to x of problem 68 for this non commuttive structure. 70. Let S = {( 1, 2,, 6 ) i [0, 9)i F ; 1 i 6, +} be the MOD complex intervl group. i. Find subgroups of infinite order in S. ii. Find ll subgroups of finite order in S Let A = { be the MOD complex intervl mtrix group. i [0, 125)i F ; 1 i 20, +} Study questions i nd ii of problem 70 for this A. 72. Cn on [0, m)i F one cn build semigroups under? Justify your clim. 73. Cn on [0, m)i F one cn build intervl pseudo ring? Justify your clim. 74. Let M = {( 1, 2,, 20 ) i [0, 27)I; 1 i 20, +} be the MOD neutrosophic intervl mtrix group. Study questions i nd ii of problem 70 for this M.

202 Suggested Problems Let B = { 2 7 i [0, 23)I; 1 i 7, +} be the MOD neutrosophic intervl mtrix group. Study questions i nd ii of problem 70 for this B. 76. Let X = { i [0, 43)I, 1 i 28, n } be the MOD neutrosophic intervl mtrix semigroup. i. Prove X is commuttive nd is of infinite order. ii. Prove X hs infinite number of zero divisors. iii. Cn X hve S-zero divisors? iv. Cn X hve S-units? v. Cn X hve finite subsemigroups which re idels? vi. Cn X hve idels of finite order? vii. Prove X hs infinite subsemigroups of infinite order which re not idels. viii. Is X S-semigroup? ix. Cn X hve S-idels? x. Enumerte other specil fetures enjoyed by X.

203 202 Multidimensionl MOD Plnes 77. Let B = { i [0, 24)I; 1 i 10, n } be the MOD neutrosophic intervl mtrix semigroup. Study questions i to x of problem 76 for this B. 78. Let V = {( 1, 2,, 9 ) where i [0, 24)I; 1 i 9, +, } be the MOD neutrosophic mtrix intervl pseudo ring. i. Prove V is of infinite order. ii. Cn V hve zero divisors? iii. Cn V hve idempotents? iv. Cn V hve S-zero divisors? v. Cn V hve S-idempotents? vi. Prove V hs no subrings of finite order. vii. Prove V hs subrings of infinite order which re not idels. viii. Prove ll pseudo idels re of infinite order. ix. Cn V hve S-idels? x. Prove V hs S-subrings which re not S-idels. xi. Find ll importnt nd interesting properties ssocited with V.

204 Suggested Problems Let V = { where i [0, 35)I; 1 i 40, +, n } be the MOD neutrosophic intervl mtrix pseudo ring. Study questions of i to xi of problem 78 for this V. 80. Let S = { i [0, 32)I; 1 i 36, n, +} be the MOD neutrosophic intervl mtrix pseudo ring. Study questions i to xi of problem 78 for this S. 81. Let M = {( 1 7 ) i [0, 24)g, g 2 = 0, 1 i 7, +} be MOD dul number group. i. Find ll finite subgroups of M. ii. Find ll infinite order subgroups of M.

205 204 Multidimensionl MOD Plnes 82. Let M 1 = { i [0, 43)g, g 2 = 0, 1 i 45, +} be the MOD dul number group. Study questions i nd ii of problem 81 for this M Let V = {( 1, 2,, 9 ) i [0, 12)g, g 2 = 0; 1 i 9, } be the MOD dul number mtrix intervl semigroup. i. Prove V is zero squre semigroup. ii. Prove every subset X in V which contins (0, 0,, 0) is MOD idel of V. iii. Prove V hs subsemigroups of order 2, 3 nd so on upto Let W = { 12 i [0, 43)g, g 2 = 0, 1 i 12, n } be the MOD dul number mtrix intervl semigroup. Study questions i to iii of problem 83 for this W.

206 Suggested Problems Let S = { i [0, 42)g, g 2 = 0, 1 i 24, n } be the MOD dul number mtrix intervl semigroup. Study questions i to iii of problem 83 for this S. 86. Let V = {( 1,, 18 ) i [0, 40)g, g 2 = 0, 1 i 18, +, n } be the MOD dul number mtrix intervl pseudo ring. i. V is zero divisor ring. ii. V hs pseudo idels of finite order s well s infinite order idels. iii. V hs infinite number of pseudo subrings ll of which re idels. iv. If W is subring of V then W is pseudo idel. Prove Let T = { 10 i [0, 21)g, g 2 = 0, 1 i 10, +, n } be the MOD dul number mtrix intervl pseudo ring. Study questions i to iv of problem 86 for this T. 88. Let M = { i [0, 16)h, h 2 = h, +, 1 i 24} be the MOD specil dul like number intervl mtrix group.

207 206 Multidimensionl MOD Plnes i. Prove M is of infinite order. ii. Prove M hs subgroups of both finite nd infinite order. 89. Let W = { i [0, 43)h, h 2 = h, 1 i 36, +} be the MOD specil dul like number intervl mtrix group. Study questions i nd ii of problem 88 for this W Let Z = { i [0, 25)h, h 2 = h; 1 i 22, +} be the MOD specil dul like number intervl mtrix group. Study questions i nd ii of problem 88 for this Z. 91. Let B = { i [0, 23)h, h 2 = h, 1 i 14, n } be the MOD dul number like mtrix intervl semigroup. i. Show B is of infinite order. ii. Cn B hve idels of finite order? iii. Cn B hve S-idels?

208 Suggested Problems 207 iv. Is B S-semigroup? v. Cn B hve subsemigroups of finite order? vi. Cn B hve zero divisors which re not S zero divisors? vii. Cn B hve S-units? viii. Show B hs non trivil idempotents. ix. Cn B hve infinite number of S-idempotents? x. Does B hs infinite order subsemigroups which re not idels? 92. Let T = { i [0, 48)h; h 2 = h; 1 i 21, n } be the MOD specil dul like number intervl mtrix semigroup. Study questions i to x of problem 91 for this T. 93. Let R = { i [0, 13)h, h 2 = h; 1 i 27, n }be the MOD specil dul like number mtrix intervl semigroup. Study questions i to x of problem 91 for this R.

209 208 Multidimensionl MOD Plnes Let M = { 3 4 i [0, 14)h, h 2 = h, 1 i 4, +, n } be the MOD specil dul like number mtrix intervl pseudo ring. i. Prove M is of infinite order. ii. Cn M hve S-zero divisors? iii. Prove M hs infinite number of zero divisors. iv. Cn M hve S idempotents? v. If I is pseudo idel of M, should I be of infinite order. vi. Show M hs finite order subrings. vii. Show M hs pseudo subrings of infinite order which re not pseudo idels. viii. Cn M hve S-unit? ix. Is M S-pseudo ring? x. Does M contin S-subrings which re not S idels? Let Z = { i [0, 15)h, h 2 = h; 1 i 8, +, n } be the MOD specil dul like number intervl mtrix pseudo ring. Study questions i to x of problem 94 for this Z.

210 Suggested Problems Let R = { i [0, 40)h, h 2 = h; 1 i 16, +, } be the MOD specil dul like number intervl mtrix pseudo ring. i. Show R is non commuttive pseudo ring. ii. Study questions i to x of problem 94 for this R. iii. Cn R hve right pseudo idels which re not pseudo left idels nd vice vers? 97. Let T = { i [0, 4)h, h 2 = h, 1 i 40, +, n } be the MOD specil dul like number mtrix intervl pseudo ring. Study questions i to x of problem 94 for this T. 98. Let V = {(x 1, x 2,, x 12 ) x i [0, 12)k, k 2 = 11k; 1 i 12, +} be the MOD specil qusi dul number mtrix intervl group. i. Show V is of infinite order. ii. Show V hs subgroups of infinite order. iii. Show V hs subgroups of finite order. iv. Obtin ny other specil feture enjoyed by V.

211 210 Multidimensionl MOD Plnes 99. Let M = { i [0, 45)k, k 2 = 44k; 1 i 54, +} be the MOD specil qusi dul number intervl mtrix group. Study questions i to iv of problem 98 for this M Let V = {( 1, 2,, 9 ) i [0, 25)k, k 2 = 24k; 1 i 9, n } be the MOD specil qusi dul number intervl mtrix semigroup. i. Prove V is of infinite order. ii. Prove V hs infinite number of zero divisors. iii. Prove V hs infinite number of S-zero divisors. iv. Prove V hs idels of infinite order. v. Cn V hve idels of finite order? vi. Cn V hve infinite order subsemigroups which re not idels of V? vii. Prove/disprove V cnnot hve S pseudo idels Let M = { i [0, 412)k, k 2 = 411k, 1 i 50, n } be the MOD specil qusi dul number intervl mtrix semigroup.

212 Suggested Problems 211 Study questions i to vii of problem 100 for this M Let W = { i [0, 24)k, k 2 = 23k, 1 i 25, } be the MOD specil qusi dul number intervl mtrix semigroup. i. Prove W is non commuttive MOD semigroup. ii. Give 2 right idels of W which re not left idels of W nd vice vers. iii. Study questions i to vii of problem 100 for this W Let M = {( 1, 2,, 9 ) i [0, 26)k, k 2 = 25k, 1 i 9, +, } be the MOD specil qusi dul number mtrix intervl pseudo ring. i. Prove M hs subrings of finite order. ii. Show M hs zero divisors. iii. Cn M infinite number of S-zero divisors? iv. Show ll MOD pseudo idels of M re of infinite order. v. Show M hs MOD pseudo subring of infinite order which re not pseudo idels. vi. Cn M hve idempotents? vii. Does M contin S-units? viii. Is M S-pseudo ring? ix. Cn M hve S-pseudo idels? x. Wht re the specil properties enjoyed by M?

213 212 Multidimensionl MOD Plnes Let S = { 12 i [0, 20)k; k 2 = 19k, 1 i 12, +, n } be the specil qusi dul number mtrix intervl pseudo ring. Study questions i to x of problem 103 for this S Let R = { where i [0, 155)k, k 2 = 154k, 1 i 45, +, n } be the specil qusi dul number mtrix intervl pseudo ring. Study questions i to x of problem 103 for this R Let P = { where i [0, 16)k, k 2 = 15k, 1 i 49, +, } be the MOD specil qusi dul number mtrix intervl MOD pseudo ring.

214 Suggested Problems 213 i. Prove P is non commuttive. ii. Study questions i to x of problem 103 for this P. iii. Give 3 exmples of right pseudo idels of P which re not left pseudo idels. iv. Give 2 exmples of left pseudo idels of P which re not right pseudo idels of P. v. Cn P hve left zero divisors which re not right zero divisors? vi. Cn P hve right units which re not left units nd vice vers? Let M 1 = { i [0, 19)k, k 2 = 18k, 1 i 9, +, } be the MOD specil qusi dul number mtrix intervl pseudo ring. Study questions i to iv of problem 106 for this M Let W = { i [0, 423)k, k 2 = 422k, 1 i 28, +, n } be the MOD specil qusi dul number mtrix intervl pseudo ring. Study questions i to x of problem 103 for this W Let (, ) be the rel line. Define MOD rel intervl trnsformtion from (, ) to [0, 17).

215 214 Multidimensionl MOD Plnes η : (, ) [0, 17). i. Is function? ii. Cn we tke bout kernel of η? iii. Is it possible to define mp from η 1 = [0, 17) (, )? iv. If η 1 usul mp? v. Cn η η 1 = η 1 η = Id (Identity)? vi. Obtin ny other specil feture enjoyed by these MOD rel intervl trnsformtion Let η : (, ) [0, 43) be the MOD rel trnsformtion. Study questions i to vi of problem 109 for this mp Let η I : ( I, I) [0, 45)I be the MOD rel intervl neutrosophic trnsformtion. Study questions i to vi of problem 109 for this η I Let η I : ( I, I) [0, 23)I be the MOD rel neutrosophic intervl trnsformtion Let Study questions i to vi of problem 109 for this η I. η i F : ( i, i) [0, 24)i F be the MOD complex modulo integer intervl trnsformtion. Study questions i to vi of problem 109 for this η i. F 114. Let η i F : ( i, i) [0, 43)i F be the MOD complex modulo integer intervl trnsformtion. Study questions i to vi of problem 109 for this ηi F Let η g : ( g, g) [0, 18)g, (g 2 = 0) be the MOD dul number intervl trnsformtion.

216 Suggested Problems 215 Study questions i to vi of problem 108 for this η g Let η g : ( g, g) [0, 29)g, (g 2 = 0) be the MOD dul number intervl trnsformtion. Study questions i to vi of problem 109 for this η g Let η h : ( h, h) [0, 144)h, (h 2 = h) be the MOD specil dul like number intervl trnsformtion. Study questions i to vi of problem 109 for this η h Let η h : ( h, h) [0, 13)h, (h 2 = h) be the MOD specil dul like number intervl trnsformtion. Study questions i to vi of problem 109 for this η h Let η k : ( k, k) [0, 248)k; (k 2 = 247k) be the MOD specil qusi dul number trnsformtion. Study questions i to vi of problem 109 for this η k Let η k : ( k, k) [0, 13)k; k 2 = 13k be the MOD specil qusi dul number trnsformtion. Study questions i to vi of problem 109 for this η k Let R n (24) be the MOD rel plne η : R R R n (24) be the MOD rel plne trnsformtion. i. Study the nturl properties of mp enjoyed by η. ii. Cn η 1 exist is it unique through not mp? iii. Wht re the difference between η 1 nd usul mp? iv. Obtin ny other specil fetures enjoyed by η nd η 1. v. Show η η 1 = η 1 η = identity.

217 216 Multidimensionl MOD Plnes 122. Let η : R R R n (47) be the MOD rel plne trnsformtion. Study questions i to v of problem 121 for this η Let η C : C C n (29) be the MOD complex plne trnsformtion. Study questions i to v of problem 121 for this η C Let η C : C C n (48) be the MOD complex plne trnsformtion. Study questions i to v of problem 121 for this η C Let η I : R I R plne trnsformtion. I n (53) be the MOD neutrosophic Study questions i to v of problem 121 for this η I Let η I : R I R plne trnsformtion. I n (448) be the MOD neutrosophic Study questions i to v of problem 121 for this η I Let η g : R(g) R n (15)g; g 2 = 0 be the MOD dul number plne trnsformtion. Study questions i to v of problem 121 for this η g Let η g : R(g) R n (43)g, g 2 = 0 be the MOD dul number plne trnsformtion. Study questions i to v of problem 121 for this η g Let η h : R(h) R n (24)h, h 2 = h be the MOD specil dul like number plne trnsformtion.

218 Suggested Problems 217 Study questions i to v of problem 121 for this η h Let η h : R(h) R n (59)h, h 2 = h be the MOD specil dul like number plne trnsformtion. Study questions i to v of problem 121 for this η h Let η k : R(k) [0, 43)k; k 2 = 42k be the MOD specil qusi dul number plne trnsformtion. Study questions i to v of problem 121 for this η k Let η k : R(k) [0, 6)k, k 2 = 5k; be the MOD specil qusi dul number plne trnsformtion. Study questions i to v of problem 121 for this η k Let M = {(R R(g) R(h) R(k)) g 2 = 0, h 2 = h nd k 2 = k, i 2 = 1, +} be the mixed multidimensionl group. Study ll interesting properties ssocited with M Study M in problem 133 if + is replced by for the semigroup Let N = {(C R(h) R(g) R(k)); h 2 = h, g 2 = 0 nd k 2 = k, } be the mixed multidimensionl semigroup. i. Prove N is commuttive monoid of infinite order. ii. Prove N hs infinite number of zero divisors. iii. Cn N hs S-zero divisors? iv. Find ll zero divisors which re not S-zero divisors. v. Cn N hve S-idempotents? vi. Does N contin infinite number of idempotents? vii. Show N hve infinite number of subsemigroups.

219 218 Multidimensionl MOD Plnes viii. Cn N hve finite order subsemigroups? ix. Cn N contin idels of finite order? x. Prove ll idels of N re of infinite order. xi. Obtin ny other interesting property enjoyed by N Let W = {(R R(g) R R(h) R(k) C C) g 2 = 0, h 2 = h, k 2 = k, i 2 = 1, } be the mixed multi dimensionl semigroup. Study questions i to xi of problem 135 for this W Let M = {(R(g) R(g) C C R(h) R(h) R(k) R(k)) g 2 = 0, h 2 = h nd k 2 = k, } be the mixed multidimensionl semigroup. Study questions i to xi of problem 135 for this M Let P = {(R(g) R(h) C C(g) C(h)) h 2 = h nd g 2 = 0, +} be the mixed multidimensionl group. P is mixed multidimensionl vector spce over R or Q. i. Is P finite dimensionl vector spce over R? ii. Find tlest 3 sets of bsis for P over R. iii. Find subspces of P over R(nd Q). iv. Define liner opertor on P. v. If V = Hom R (P, P) find dimension of V over R. vi. Is V P? vii. Obtin ny other interesting property ssocited with P. viii. Give liner trnsformtion in which Ker is non empty subspce of P. ix. Give nontrivil liner trnsformtion which is one to one. x. Obtin ny other property ssocited with V. xi. Is projection from W = {(R(g) {0} {0} {0} {0})} (be subspce of P) to P well defined. xii. Write P s direct sum of subspces.

220 Suggested Problems Let S = {(R(h) C(h) R C R(g) R(k)) g 2 = 0, h 2 = h nd k 2 = k, i 2 = 1, +} be the mixed multidimensionl vector spce over R (or Q). Study questions i to xii of problem 138 for this S over R (nd Q) Let M = {(R(h) R R(k) C C(h) R(g)), +, } be the mixed multidimensionl ring. i. Prove M is commuttive. ii. Prove M hs infinite number of zero divisors. iii. Cn M hve S-zero divisors? iv. Is M S-ring? v. Cn M hve S-idels? vi. Does M hve idels which re not S idels? vii. Cn M hve S-idempotents? viii. Cn M hve subrings of finite order? ix. Obtin ny other specil property enjoyed by M. x. Does M contin S-subrings? 141. Let S = {(C(g) C(h) R I R(g) C I R(h)) g 2 = 0, h 2 = h, I 2 = I, i 2 = 1, +, } be the mixed multidimensionl ring. Study questions i to x of problem 140 for this S Let S be s in problem 141. S is mixed multidimensionl liner lgebr over R. i. Wht is dimension of S over R? ii. Is S finite dimensionl? iii. Find subspces of S over R so tht S cn be written s direct sum. iv. Find Hom(S, S) = V. v. Wht is the lgebric structure enjoyed by V? vi. Find Hom(S, R) = W. vii. Find the lgebric structure enjoyed by W. viii. Is W S?

221 220 Multidimensionl MOD Plnes 143. Let V = {(R n (m) R n (m) R n (m)) m = 8, } be the MOD multidimensionl semigroup. i. Cn V hve subgroups of finite order? ii. Cn idels of V be of finite order? iii. Cn V hve S-units? iv. Cn V hve S-idempotents? v. Cn V hve S-zero divisors? vi. Is V S-semigroup? vii. Cn V hve S-idels? viii. Does V contin S-subsemigroups which re not S-idels? ix. Obtin ny other specil feture enjoyed by this MOD multidimensionl semigroup Let S = {(C n (10) R n (5) R n (12)g R n (11)) g 2 = 0, } be MOD mixed multidimensionl semigroup. Study properties i to ix of problem 143 for this S Let B = {(R n (7)g R n (7)g R n (7)g R n (7)g) g 2 = 0, } be the MOD multidimensionl semigroup. Study properties i to ix of problem 143 for this B Let 2 P = {(C n (10) C n (10) C n (10) C n (10)); i F = 9, } be the MOD multidimensionl complex modulo integer semigroup. Study questions i to ix of problem 143 for this P Let M = {(R n (9)h R n (9)h R n (9)h) h 2 = h, } be the MOD multidimensionl specil dul like number spce semigroup. Study questions i to ix of problem 143 for this M.

222 Suggested Problems Let S = {(C n (10) R n (5) R n (7)g R n (5)h); } be the MOD mixed multidimensionl spce semigroup. Study questions i to ix of problem 143 for this S Let P = {(R n (10)g R n (11)h R n (12)k C n (10) C n (11)), g 2 = 0, h 2 = h, k 2 = 11k, } be the MOD mixed multidimensionl spce semigroup. Study questions i to ix of problem 143 for this P Let M = {(R n (5) R n (5)h C n (5) R n (5)k) h 2 = h, k 2 = 4k, +} be the MOD mixed multidimensionl spce group. M is MOD mixed multidimensionl vector spce over Z 5. i. Wht is the dimension of M over Z 5? ii. Is M n infinite dimensionl mixed multidimensionl vector spce over Z 5? iii. Find subspces of V. iv. Does there exist subspce of finite dimension over Z 5? v. Find Hom(V, V) = W. vi. Wht is the lgebric property enjoyed by W? vii. Obtin ny other specil fetures ssocited with W Let S = {(C n (7) C n (5) C n (10) R n (7) R n (5)) +, } be the MOD mixed multidimensionl spce pseudo ring. i. S is S-ring. ii. Prove S hs infinite number of zero divisors. iii. Prove S hs only finite number of units. iv. Cn S hve S-units? v. Does S contin S-idels? vi. Cn S hve idels of finite order? vii. Prove S hs S-subrings? viii. Cn S hve S-idempotents?

223 222 Multidimensionl MOD Plnes ix. Let I be n idel of S-find S/I. x. Obtin ny other property ssocited with S Let M = {(R n (7) R n (5)g R n (12) C n (8) R n (8)h) g 2 = 0, h 2 2 = h, i F = 7, +, } be the MOD mixed multidimensionl spce pseudo ring. Study questions i to x of problem 151 for this M Let W = {(C n (7) C n (7) C n (7) C n (7)), +, } be the MOD multidimensionl spce pseudo ring. Study questions i to x of problem 151 for this W Let B = {(C n (10) R n (7) R n (5)g R n (12) R n (17)h) h 2 = h, g 2 2 = 0, i F = 9, +, } be the MOD mixed multidimensionl spce pseudo ring. Study questions i to x of problem 151 for this B Let V = {(C n (11) R n (11) R n (11)h R n (11)g R n (11)k) k 2 = 10k, g 2 = 0, h 2 = h, +, } be the MOD mixed multidimensionl pseudo ring. i. Prove V is MOD mixed multidimensionl pseudo liner lgebr over Z 11. ii. Prove V is S-MOD mixed multidimensionl pseudo liner lgebr over [0, 11). iii. Is V finite dimensionl over [0, 11)? iv. Obtin ny other interesting property bout V Mention some of the specil fetures enjoyed by MOD multidimensionl pseudo ring Distinguish between the MOD mixed multidimensionl pseudo ring nd MOD multidimensionl pseudo ring.

224 Suggested Problems Mention ll the innovting properties enjoyed by MOD multidimensionl pseudo liner lgebrs over [0, m) Enumerte the specil fetures enjoyed by MOD mixed multidimensionl pseudo liner lgebrs over Z p. (p prime) If Z p in problem 159 is replced by [0, p) study the relted structures.

225 FURTHER READING 1. Herstein., I.N., Topics in Algebr, Wiley Estern Limited, (1975). 2. Lng, S., Algebr, Addison Wesley, (1967). 3. Smrndche, Florentin, Definitions Derived from Neutrosophics, In Proceedings of the First Interntionl Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probbility nd Sttistics, University of New Mexico, Gllup, 1-3 December (2001). 4. Smrndche, Florentin, Neutrosophic Logic Generliztion of the Intuitionistic Fuzzy Logic, Specil Session on Intuitionistic Fuzzy Sets nd Relted Concepts, Interntionl EUSFLAT Conference, Zittu, Germny, September Vsnth Kndsmy, W.B., On Muti differentition of deciml nd Frctionl Clculus, The Mthemtics Eduction, 38 (1994) Vsnth Kndsmy, W.B., On deciml nd Frctionl Clculus, Ultr Sci. of Phy. Sci., 6 (1994)

226 Further Reding Vsnth Kndsmy, W.B., Muti Fuzzy differentition in Fuzzy Polynomil rings, J. of Inst. Mth nd Comp. Sci., 9 (1996) Vsnth Kndsmy, W.B., Some Applictions of the Deciml Derivtives, The Mthemtics Eduction, 34, (2000) Vsnth Kndsmy, W.B., On Prtil Differentition of Deciml nd Frctionl clculus, J. of Inst. Mth nd Comp. Sci., 13 (2000) Vsnth Kndsmy, W.B., On Fuzzy Clculus over fuzzy Polynomil rings, Vikrm Mthemticl Journl, 22 (2002) Vsnth Kndsmy, W. B. nd Smrndche, F., N-lgebric structures nd S-N-lgebric structures, Hexis, Phoenix, Arizon, (2005). 12. Vsnth Kndsmy, W. B. nd Smrndche, F., Neutrosophic lgebric structures nd neutrosophic N-lgebric structures, Hexis, Phoenix, Arizon, (2006). 13. Vsnth Kndsmy, W. B. nd Smrndche, F., Smrndche Neutrosophic lgebric structures, Hexis, Phoenix, Arizon, (2006). 14. Vsnth Kndsmy, W.B., nd Smrndche, F., Fuzzy Intervl Mtrices, Neutrosophic Intervl Mtrices nd their Applictions, Hexis, Phoenix, (2006). 15. Vsnth Kndsmy, W.B. nd Smrndche, F., Finite Neutrosophic Complex Numbers, Zip Publishing, Ohio, (2011). 16. Vsnth Kndsmy, W.B. nd Smrndche, F., Dul Numbers, Zip Publishing, Ohio, (2012). 17. Vsnth Kndsmy, W.B. nd Smrndche, F., Specil dul like numbers nd lttices, Zip Publishing, Ohio, (2012).

227 226 Multidimensionl MOD plnes 18. Vsnth Kndsmy, W.B. nd Smrndche, F., Specil qusi dul numbers nd Groupoids, Zip Publishing, Ohio, (2012). 19. Vsnth Kndsmy, W.B. nd Smrndche, F., Algebric Structures using Subsets, Eductionl Publisher Inc, Ohio, (2013). 20. Vsnth Kndsmy, W.B. nd Smrndche, F., Algebric Structures using [0, n), Eductionl Publisher Inc, Ohio, (2013). 21. Vsnth Kndsmy, W.B. nd Smrndche, F., Algebric Structures on the fuzzy intervl [0, 1), Eductionl Publisher Inc, Ohio, (2014). 22. Vsnth Kndsmy, W.B. nd Smrndche, F., Algebric Structures on Fuzzy Unit squres nd Neturosophic unit squre, Eductionl Publisher Inc, Ohio, (2014). 23. Vsnth Kndsmy, W.B. nd Smrndche, F., Algebric Structures on Rel nd Neturosophic squre, Eductionl Publisher Inc, Ohio, (2014). 24. Vsnth Kndsmy, W.B., Ilnthenrl, K., nd Smrndche, F., MOD plnes, EuropNov, (2015). 25. Vsnth Kndsmy, W.B., Ilnthenrl, K., nd Smrndche, F., MOD Functions, EuropNov, (2015).

228 INDEX D Deficit MOD pseudo intervl mp, 8-11 Demi MOD intervl, 8-10 Demi MOD pseudo intervl trnsformtion, 8-11 M Mixed multi n-dimensionl specil spce, MOD column rel mtrix, 43-6 MOD complex modulo integer intervl mtrix group, 45-7 MOD complex modulo integer intervl mtrix, 45-9 MOD complex modulo integer intervl subgroup, 46-8 MOD complex modulo integer intervl trnsformtion, 90-2 MOD complex plne, MOD complex rtionl number group, 15-6 MOD dul number idel, 24-6 MOD dul number intervl group, 23-5 MOD dul number intervl mtrix semigroup, 57-9

229 228 Multidimensionl MOD Plnes MOD dul number intervl trnsformtion, 22-3 MOD dul number intervl, 20-3 MOD dul number mtrix intervl idel, 70-2 MOD dul number mtrix intervl ring, 70-2 MOD dul number plne trnsformtion, MOD dul number ring, 26-7 MOD dul number semigroup, 24-5 MOD dul number subgroup, 23-5 MOD dul number subring, 26-7 MOD dul number subsemigroup, 24-5 MOD dul number zero squre ring, 28-9 MOD dul number zero squre semigroup, 24-5 MOD dul number zero squre subrings, 28-9 MOD dul specil pseudo intervl, 9-10 MOD Euclid plne, MOD finite complex modulo integer plne, MOD finite complex modulo integer trnsformtion, MOD integer semi complex number intervl, MOD intervl complex modulo integer semigroup, 53-5 MOD intervl dul number subring, 70-2 MOD mixed multi dimensionl modulo integer spce, MOD mixed multi dimensionl neutrosophic group, MOD mixed multi dimensionl neutrosophic spce, MOD mixed multi dimensionl neutrosophic semigroup, MOD mixed multi dimensionl specil dul like number semigroup, MOD mixed rel multi dimensionl spce, MOD modulo integer intervl dul number mtrix group, MOD modulo integer intervl dul number mtrix, MOD modulo integer intervl neutrosophic mtrix group, 47-9 MOD modulo integer intervl neutrosophic mtrix, 47-9 MOD multi dimensionl complex modulo integer pseudo ring, MOD multi dimensionl complex modulo integer semigroup, MOD multi dimensionl complex spce, MOD multi dimensionl dul number pseudo ring, MOD multi dimensionl dul number spce, MOD multi dimensionl neutrosophic pseudo ring,

230 MOD multi dimensionl neutrosophic semigroup, MOD multi dimensionl rel neutrosophic group, 146 MOD multi dimensionl specil dul like number pseudo ring, MOD multi dimensionl specil dul like number spce, MOD neutrosophic intervl mtrix pseudo ring, MOD neutrosophic intervl mtrix semigroup, 54-6 MOD neutrosophic intervl trnsformtion, 90-3 MOD neutrosophic mtrix intervl pseudo idels, MOD neutrosophic mixed multi dimensionl pseudo ring, MOD neutrosophic mixed multi dimensionl pseudo subring, MOD rel intervl mtrix pseudo idels, 62-8 MOD rel intervl mtrix pseudo ring, 62-8 MOD rel intervl mtrix subring, 62-8 MOD rel intervl rel squre mtrix, 44-7 MOD rel intervl trnsformtion, 83-5 MOD rel intervl, 17-9 MOD rel mixed multi dimensionl pseudo ring, MOD rel neutrosophic intervl trnsformtion, MOD rel number intervl mtrix semigroup, 52-4 MOD rel plne, MOD rel trnsformtion, 17-9 MOD row intervl rel mtrix group, 43-6 MOD row rel mtrix, 43-6 MOD specil dul like intervl number pseudo ring, 72-5 MOD specil dul like number group, 31-3 MOD specil dul like number intervl mtrix group, 50-3 MOD specil dul like number intervl mtrix semigroup, 58-9 MOD specil dul like number intervl pseudo idel, 72-5 MOD specil dul like number intervl pseudo subring, 72-5 MOD specil dul like number intervl subring, 72-5 MOD specil dul like number intervl trnsformtion, 30-1 MOD specil dul like number intervl, 29-3 MOD specil dul like number pseudo ring, 33-5 MOD specil dul like number pseudo subring, 33-6 MOD specil dul like number semigroup, 31-3 MOD specil dul like number subgroups, 31-3 Index 229

231 230 Multidimensionl MOD Plnes MOD specil dul like number subsemigroup, 32-4 MOD specil dul number specil dul like trnsformtion, MOD specil dul qusi dul number trnsformtion, 37-9 MOD specil multi dimensionl qusi dul number subsemigroup, MOD specil qusi dul number idels, 41-3 MOD specil qusi dul number intervl group, 38-9 MOD specil qusi dul number intervl mtrix group, 51-2 MOD specil qusi dul number intervl mtrix pseudo idel, 76-9 MOD specil qusi dul number intervl mtrix pseudo ring, 74-6 MOD specil qusi dul number intervl mtrix semigroup, MOD specil qusi dul number intervl mtrix subring, 76-9 MOD specil qusi dul number intervl mtrix, 51-2 MOD specil qusi dul number intervl semigroup, 40-2 MOD specil qusi dul number intervl subgroup, 38-9 MOD specil qusi dul number intervl subsemigroup, 40-3 MOD specil qusi dul number intervl trnsformtion, MOD specil qusi dul number intervl, 36-7 MOD specil qusi dul number multi dimensionl group, MOD specil qusi dul number multi dimensionl pseudo ring, MOD specil qusi dul number multi dimensionl semigroup, MOD specil qusi dul number multi dimensionl spce, MOD specil qusi dul number ring, 41-3 MOD specil qusi dul number subring, 41-3 MOD specil qusi dul number trnsformtion, MOD specil dul like number intervl mtrix, 50-3 MOD zero squre dul number semigroup, 25-6 Multi dimensionl rel plne, Multi n-dimension complex plne, Multi n-dimensionl dul number plne, Multi n-dimensionl specil dul like number plne, Multi n-dimensionl specil qusi dul number plne, 132-6

232 Index 231 P Pseudo mpping, 8-10 S Semi MOD complex modulo integer intervl, Semi MOD pseudo intervl mp, 8-11, Semi MOD pseudo intervl trnsformtion, 8-11 Semi MOD pseudo intervls, 7-9 Semi pseudo MOD intervl mp, 8-11 Smrndche dul number vector spce, S-pseudo multi dimensionl dul number vector spce, S-pseudo multi dimensionl mixed pseudo liner lgebr, 167-9

233 ABOUT THE AUTHORS Dr.W.B.Vsnth Kndsmy is Professor in the Deprtment of Mthemtics, Indin Institute of Technology Mdrs, Chenni. In the pst decde she hs guided 13 Ph.D. scholrs in the different fields of non-ssocitive lgebrs, lgebric coding theory, trnsporttion theory, fuzzy groups, nd pplictions of fuzzy theory of the problems fced in chemicl industries nd cement industries. She hs to her credit 694 reserch ppers. She hs guided over 100 M.Sc. nd M.Tech. projects. She hs worked in collbortion projects with the Indin Spce Reserch Orgniztion nd with the Tmil Ndu Stte AIDS Control Society. She is presently working on reserch project funded by the Bord of Reserch in Nucler Sciences, Government of Indi. This is her 106 th book. On Indi's 60th Independence Dy, Dr.Vsnth ws conferred the Klpn Chwl Awrd for Courge nd Dring Enterprise by the Stte Government of Tmil Ndu in recognition of her sustined fight for socil justice in the Indin Institute of Technology (IIT) Mdrs nd for her contribution to mthemtics. The wrd, instituted in the memory of Indin-Americn stronut Klpn Chwl who died bord Spce Shuttle Columbi, crried csh prize of five lkh rupees (the highest prize-money for ny Indin wrd) nd gold medl. She cn be contcted t vsnthkndsmy@gmil.com Web Site: or Dr. K. Ilnthenrl is the editor of The Mths Tiger, Qurterly Journl of Mths. She cn be contcted t ilnthenrl@gmil.com Dr. Florentin Smrndche is Professor of Mthemtics t the University of New Mexico in USA. He published over 75 books nd 200 rticles nd notes in mthemtics, physics, philosophy, psychology, rebus, literture. In mthemtics his reserch is in number theory, non-eucliden geometry, synthetic geometry, lgebric structures, sttistics, neutrosophic logic nd set (generliztions of fuzzy logic nd set respectively), neutrosophic probbility (generliztion of clssicl nd imprecise probbility). Also, smll contributions to nucler nd prticle physics, informtion fusion, neutrosophy ( generliztion of dilectics), lw of senstions nd stimuli, etc. He got the 2010 Telesio-Glilei Acdemy of Science Gold Medl, Adjunct Professor (equivlent to Doctor Honoris Cus) of Beijing Jiotong University in 2011, nd 2011 Romnin Acdemy Awrd for Technicl Science (the highest in the country). Dr. W. B. Vsnth Kndsmy nd Dr. Florentin Smrndche got the 2012 New Mexico-Arizon nd 2011 New Mexico Book Awrd for Algebric Structures. He cn be contcted t smrnd@unm.edu

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