INTERVAL SEMIGROUPS. W. B. Vasantha Kandasamy Florentin Smarandache

Transcription

1 Interval Semgroups - Cover.pdf:Layout 1 1/20/ :04 AM Page 1

2 INTERVAL SEMIGROUPS W. B. Vasantha Kandasamy Florentn Smarandache KAPPA & OMEGA Glendale 2011

3 Ths book can be ordered n a paper bound reprnt from: Books on Demand ProQuest Informaton & Learnng (Unversty of Mcroflm Internatonal) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI , USA Tel.: (Customer Servce) Copyrght 2011 by Kappa & Omega and the Authors 6744 W. Northvew Ave. Glendale, AZ 85303, USA Peer revewers: Prof. Cataln Barbu, Vasle Alecsandr College, Bacau, Romana Prof. Mhàly Bencze, Department of Mathematcs Áprly Lajos College, Braov, Romana Dr. Fu Yuhua, , Lufangbel Lufang Street, Chaoyang dstrct, Bejng, P. R. Chna Many books can be downloaded from the followng Dgtal Lbrary of Scence: ISBN-10: ISBN-13: EAN: Prnted n the Unted States of Amerca 2

4 CONTENTS Preface 5 Dedcaton 6 Chapter One INTRODUCTION 7 Chapter Two INTERVAL SEMIGROUPS 9 Chapter Three INTERVAL POLYNOMIAL SEMIGROUPS 37 Chapter Four SPECIAL INTERVAL SYMMETRIC SEMIGROUPS 47 Chapter Fve NEUTROSOPHIC INTERVAL SEMIGROUPS 61 3

5 Chapter Sx NEUTROSOPHIC INTERVAL MATRIX SEMIGROUPS AND FUZZY INTERVAL SEMIGROUPS Pure Neutrosophc Interval Matrx Semgroups Neutrosophc Interval Polynomal Semgroups Fuzzy Interval Semgroups 118 Chapter Seven APPLICATION OF INTERVAL SEMIGROUPS 129 Chapter Eght SUGGESTED PROBLEMS 131 FURTHER READING 159 INDEX 161 ABOUT THE AUTHORS 165 4

6 PREFACE In ths book we ntroduce the noton of nterval semgroups usng ntervals of the form [0, a], a s real. Several types of nterval semgroups lke fuzzy nterval semgroups, nterval symmetrc semgroups, specal symmetrc nterval semgroups, nterval matrx semgroups and nterval polynomal semgroups are defned and dscussed. Ths book has eght chapters. The man feature of ths book s that we suggest 241 problems n the eghth chapter. In ths book the authors have defned 29 new concepts and llustrates them wth 231 examples. Certanly ths wll fnd several applcatons. The authors deeply acknowledge Dr. Kandasamy for the proof readng and Meena and Kama for the formattng and desgnng of the book. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 5

7 ~ DEDICATED TO ~ Ayyankal Ayyankal ( ) was the frst leader of Dalts from Kerala. He ntated several reforms to emancpate the lves of the Dalts. Ayyankal organzed Dalts and fought aganst the dscrmnatons done to Dalts and through hs efforts he got the rght to educaton, rght to walk on the publc roads and dalt women were allowed to cover ther nakedness n publc. He spearheaded movements aganst castesm. 6

8 Chapter One INTRODUCTION We n ths book make use of specal type of ntervals to buld nterval semgroups, nterval row matrx semgroups, nterval column matrx semgroups and nterval matrx semgroups. We also ntroduce and study the Smarandache analogue of them. The new noton of nterval symmetrc semgroups and specal nterval symmetrc semgroups are defned and studed. For more about symmetrc semgroups and ther Smarandache analogue concepts please refer [9]. The classcal theorems for fnte groups lke Lagrange theorem, Cauchy theorem and Sylow theorem are ntroduced n a specal way and analyzed. Only under specal condtons we see the noton of these classcal theorems for fnte groups can be extended nterval semgroups. The authors also ntroduce the noton of neutrosophc ntervals and fuzzy ntervals and study them n the context of nterval semgroups. I denotes the ndetermnate or ndermnancy where I 2 = I and I + I = 2I, I + I + I = 3I and so 7

9 on. For more about neutrosophy, neutrosophc ntervals please refer [1, 3, 6-8]. Study of specal elements lke nterval zerodvsors, nterval dempotents, nterval unts, nterval nlpotents are studed and ther Smarandache analogue ntroduced [9]. 8

10 Chapter Two INTERVAL SEMIGROUPS In ths chapter we for the frst tme ntroduce the noton of nterval semgroups and descrbe a few of ther propertes assocated wth them. We see n general several of the classcal theorems are not true n general case of semgroups. Frst we proceed on to gve some notatons essental to develop these new structures. I (Z n ) = {[0, a m ] a m Z n }, I(Z + {0}) = {[0, a] a Z + {0}}, I(Q + {0}) = {[0, a] a Q + {0}}, I(R + {0}) = {[0, a] a R + {0}} and I(C + {0}) = {[0, a] a C + {0}}. DEFINITION 2.1: Let S = {[0, a ] a Z n ; +} S s a semgroup under addton modulo n. S s defned as the nterval semgroup under addton modulo n. We wll frst llustrate ths by some smple examples. 9

11 Example 2.1: Let S = {[0, a ] a Z 6 }, under addtons s an nterval semgroup. We see S s of fnte order and order of S s sx. Example 2.2: Let S = {[0, a ] a Z 12 } be an nterval semgroup under addton modulo 12. Ths s also a nterval semgroup of fnte order. Now we can defne nterval semgroup under addton usng Z + {0}, Q + {0}, R + {0} and C + {0}. All these nterval semgroups are of nfnte order. We wll llustrate these stuatons by some examples. Example 2.3: Let S = {[0, a ] a Z + {0}}; S s an nterval semgroup under addton. Clearly S s of nfnte order. Example 2.4: Let S = {[0, a ] a Q + {0}}; S s an nterval semgroup under addton. Clearly S s of nfnte order. Example 2.5: Let S = {[0, a ] a R + {0}}; S s an nterval semgroup under addton and s of nfnte order. Example 2.6: Let S = {[0, a ] a C + {0}}; S s an nterval semgroup under addton and s of nfnte order. Thus we have seen examples of nterval semgroups under addton, these are known as basc nterval semgroups under addton. We wll now defne polynomal nterval semgroups and matrx nterval semgroups defned usng basc nterval semgroups and then defne ther substructures. DEFINITION 2.2: Let S = {([0, a 1 ], [0, a 2 ],, [0, a n ]) a Z n }; S under component wse addton s an nterval semgroup known as the row matrx nterval semgroup. We can n the defnton replace Z n by Z + {0} or R + {0} or Q + {0} or C + {0}. 10

12 We wll llustrate these by some examples. Example 2.7: Let S = {([0, a 1 ], [0, a 2 ], [0, a 3 ], [0, a 4 ], [0, a 5 ]) a Z 12 ; 1 5} s a row matrx nterval semgroup under addton. Example 2.8: Let P = {([0, a 1 ], [0, a 2 ], [0, a 3 ], [0, a 4 ], [0, a 5 ], [0, a 6 ]) a Z + {0}; 1 6} s a row matrx nterval semgroup under addton. Clearly P s of nfnte order. Example 2.9: Let S = {([0, a 1 ], [0, a 2 ], [0, a 3 ],, [0, a 12 ]) / a Q + {0}; 1 12}; S s a row matrx nterval semgroup under addton and s of nfnte order. Example 2.10: Let S = {([0, a 1 ], [0, a 2 ], [0, a 3 ],, [0, a 15 ]) / a R + {0}; 1 15} be a row matrx nterval semgroup under addton; T s of nfnte order. Example 2.11: Let G = {([0, a 1 ], [0, a 2 ]) a C + {0}; 1 2}; be a row matrx nterval semgroup of nfnte order. Now we proceed onto defne column matrx nterval semgroup. DEFINITION 2.3: Let S = 1 ] 2 ] a Z m; 3 ], 1n n ] S under addton modulo m s a semgroup defned as the column nterval matrx semgroup under addton. We can replace Z n s defnton 2.3 by Z + {0} or Q + {0}, R + {0} or C + {0} and get column nterval matrx semgroups under addton. 11

13 We wll llustrate these stuatons by some examples. Example 2.12: Let S = 1] 2] 3] az 5;1 5 4] 5] be a column nterval matrx semgroup under addton. S s of fnte order. Example 2.13: Let S = 1] 2 ] 3] a Z {0}; ] be a column nterval matrx semgroup under addton. Example 2.14: Let S = 1] 2 ] 3] a Q {0};1 10, 10 ] be a column nterval semgroup under addton of nfnte order. 12

14 Example 2.15: Let 1] 2 ] S = 3] a R {0};1 6 6 ] be a column nterval semgroup under addton of nfnte order. Example 2.16: Let P = 1] ] 4 ] a C {0}; ] be a column nterval semgroup under addton of nfnte order. Now we wll defne matrx nterval semgroup. DEFINITION 2.4: Let S = {m n nterval matrces wth entres from I(Z n )} be a m n matrx nterval semgroup under addton. We can replace I (Z n ) n defnton 2.4 by I (Z + {0}) or I (R + {0}) or I(Q + {0}) or I(C + {0}) and get m n nterval matrx semgroups under addton. We wll llustrate all these by some examples. Example 2.17: Let 1] 2] S = 3] 4] a Z 10;1 6 5] 6] 13

15 be a 3 2 nterval matrx semgroup under matrx addton modulo 10 of fnte order. Example 2.18: Let S = 1] 2] 3] 4] 5] 6] a Z 42;1 9 7] 8] 9] be a 3 3 nterval square matrx semgroup of fnte order under nterval matrx addton modulo 42. Example 2.19: Let S = 1] 2] 3] ] ] ] 22 ] 23 ] 24 ] a Q {0}; be a 8 3 matrx nterval semgroup under addton of nfnte order. Example 2.20: Let P = 1] 4] ] ] 13 ] 16 ] a R {0}; ] 12] be a 4 4 square matrx nterval semgroup of nfnte order. 14

16 DEFINITION 2.5: Let S be the matrx nterval semgroup under addton. Let M S (M a proper subset of S), f M tself s a matrx nterval semgroup under addton then we defne M to be a matrx nterval subsemgroup of S. We wll llustrate ths stuaton by some examples. Example 2.21: Let S = 1] 2] 3] 4] a Z 5;14 be a square matrx nterval semgroup under addton. P = ] ] ] ] a Z S; P s a square matrx nterval subsemgroup of S. Example 2.22: Let S = 1] 2] 3] 4] a Q {0};1 6 5] 6] be a 3 2 matrx nterval semgroup under addton. P = ] ] ] ] a Z {0} ] ] S s a 3 2 matrx nterval subsemgroup of S under addton. 15

17 Example 2.23: Let M = 1] ] 12 ] a R {0} 2 be a 12 1 column matrx nterval semgroup under addton. Choose S = ] ] ar {0} ] M; S s a 12 1 column matrx nterval subsemgroup of M. Example 2.24: Let M = {([0, a 1 ], [0, a 2 ],, [0, a 19 ]) a C + {0}; 1 19} be a 1 19 row matrx nterval semgroup. Choose W = {([0, a], [0, a],, [0, a]) a R + {0}} M; W s a 1 19 row matrx nterval subsemgroup of M. We can defne deals as n case of usual semgroups. DEFINITION 2.6: Let S be a matrx nterval semgroup under addton. I a proper subset of S. I s sad to be a matrx nterval deal of the semgroup S f (a) I s a matrx nterval subsemgroup of S. (b) for each s S and a I. a + s and s + a are n I. The noton of left and rght deal n a matrx nterval semgroup can also be defned as a matter of routne. The followng theorem s obvous and the reader s expected to prove. 16

18 THEOREM 2.1: Let S be a matrx nterval semgroup. Every deal I of S s a matrx nterval subsemgroup of S but a matrx nterval subsemgroup n general s not a matrx nterval deal of S. We wll llustrate ths by some smple examples. Example 2.25: Let S be a 1 5 row matrx nterval semgroup under addton wth entres from 3Z + {0}. Choose I = {([0, a 1 ], [0, a 2 ], [0, a 3 ], [0, a 4 ], [0, a 5 ]) a 6Z + {0}} S s not a 1 5 matrx nterval deal of S. We see some matrx nterval subsemgroups of S are not n general matrx nterval deals. For take J = {([0, a], [0, a],, [0, a]) a 9Z + {0}} S. We see J s not a 1 5 matrx nterval deal, J s only a 1 5 matrx nterval subsemgroup of S under addton. Example 2.26: Let S = {([0, a 1 ], [0, a 2 ],, [0, a 10 ]) a Z 12 } be a 1 10 row nterval matrx semgroup. Consder any 1 10 row nterval matrx subsemgroup, we see t cannot be an deal. For example take V = {([0, a 1 ], [0, a 2 ],, [0, a 10 ]) a {0, 6} Z 12 } S; V s a 1 10 row nterval matrx subsemgroup of S but s not an deal. For take 2 Z 12 we see V + 2 V for V + 2 = {([0, a 1 ], [0, a 2 ],, [0, a 10 ]) a {0, 6, 8, 2}} V. Thus we see V has no 1 10 row nterval matrx deal. We see t s dffcult to get deals n case of row matrx nterval semgroups under addton, but however we have deals n case of row matrx nterval semgroups under multplcaton. We have a class of nterval matrx semgroups under addton whch have no deals. We wll call those nterval matrx semgroups whch have no subsemgroups as smple and those wll have no deals. Those nterval matrx semgroups whch has subsemgroups but no deals as doubly smple. Example 2.27: Let S = {([0, a 1 ], [0, a 2 ],, [0, a 9 ]) a Z 5 ; 1 9} be a nterval matrx semgroup under addton, S has no 17

19 nterval matrx subsemgroups as well as S has no nterval matrx deals. Infact S s a doubly smple nterval matrx as S has no nterval matrx deals and subsemgroups. Now we proceed onto defne nterval matrx semgroup under multplcaton. DEFINITION 2.7: Let S be a nterval matrx semgroup under multplcaton usng I(Z n ) or I (Z + {0}) or I (Q + {0}) or I(R + {0}). We wll llustrate ths stuaton by examples. Example 2.28: Let V = {([0, a 1 ], [0, a 2 ],, [0, a 9 ]) a Z 12 ; 1 9}; V s an nterval matrx semgroup under multplcaton. Clearly V s a fnte nterval matrx semgroup. Example 2.29: Let V = {([0, a 1 ], [0, a 2 ],, [0, a 10 ]) a Z + {0}; 1 10} be an nterval row matrx semgroup under multplcaton. Clearly V s of nfnte order. Example 2.30: Let V = {([0, a 1 ], [0, a 2 ],, [0, a 12 ]) a Q + {0}; 1 12} be an nterval row matrx semgroup under multplcaton. We can defne subsemgroups and deals n case of these semgroups. DEFINITION 2.8: Let V = {([0, a 1 ], [0, a 2 ],, [0, a n ]) / a Q + {0} (or Z n or R + {0} or Z + {0}} be a row nterval matrx semgroup under multplcaton. Let P = {([0, a 1 ], [0, a 2 ],, [0, a n ])} V, f P under the operatons of V s matrx semgroup then we call P to be a row nterval matrx subsemgroup of V under multplcaton. We llustrate ths by some examples. Example 2.31: Let V = {([0, a 1 ], [0, a 2 ],, [0, a 8 ]) a Z 24 } be a row matrx nterval semgroup under multplcaton. 18

20 Choose P = {([0, a 1 ], [0, a 2 ],, [0, a 8 ]) a {0, 2, 4, 6,, 22}; 1 8} V; V s a row matrx nterval subsemgroup of V. Example 2.32: Let V = {([0, a], [0, a], [0, a], [0, a]) a Z 5 \ {0}} be a row matrx nterval semgroup under multplcaton. Clearly V has a row matrx nterval subsemgroup. In vew of ths we say a row matrx nterval semgroup s smple f t has no proper row matrx nterval subsemgroups. We have a large class of smple row matrx nterval semgroups. We say proper f the row nterval matrx semgroup s not {([0, 0],, [0, 0])} or {([0, 1], [0, 1],, [0, 1])}. These two semgroups wll be known as mproper row matrx nterval subsemgroup or trval row matrx nterval subsemgroup. Example 2.33: Let V = {([0, a 1 ], [0, a 2 ],, [0, a 9 ]) a Z 240 ; 1 9} be a row matrx nterval semgroup under multplcaton. P = {([0, a 1 ], [0, a 2 ],, [0, a 9 ]) a {0, 10, 20, 30, 40,, 230}; 1 9} V s a row matrx nterval subsemgroup of V. THEOREM 2.2: Let V = {([0, a], [0, a],, [0, a]) a Z p \ {0}} p a prme be a 1 n row nterval matrx semgroup under multplcaton. V s not a smple 1 n row nterval matrx semgroup. The proof s left as an exercse for the reader. THEOREM 2.3: Let V = {([0, a 1 ], [0, a 2 ],, [0, a n ]) a Z n ; n not a prme} be a 1 n row nterval matrx semgroup. V s not a smple 1 n row nterval matrx semgroup (V has proper subsemgroup). Now we can defne deals of a 1 n row nterval matrx semgroup as follows. 19

21 DEFINITION 2.9: Let V be a 1 n row nterval matrx semgroup. P V; be a proper subsemgroup. We say P s a 1 n row nterval matrx deal of V f for all p P and v V, pv and vp are n P. We wll llustrate ths stuaton by some examples. Example 2.34: Let V = {([0, a 1 ], [0, a 2 ], [0, a 3 ], [0, a 4 ], [0, a 5 ]) / a Z 12 ; 1 5} be a 1 5 row nterval matrx semgroup. Choose P = {([0, a 1 ], [0, a 2 ],, [0, a 5 ]) a {0, 2, 4, 6, 8, 10} Z 12 } V to be a row nterval matrx subsemgroup of V. It s easly verfed V s a row nterval matrx deal of V. Example 2.35: Let V = {([0, a 1 ], [0, a 2 ],, [0, a 9 ]) a Z + {0}; 1 9} be a 1 9 row matrx nterval semgroups. Choose P = {([0, a 1 ], [0, a 2 ],, [0, a 9 ]) a 5Z + {0}; 1 9} V s a 1 9 row nterval matrx deal of V. Example 2.36: Let V = {([0, a 1 ], [0, a 2 ],, [0, a n ]) a Z p ; 1 n} be a 1 n row nterval matrx semgroup; p a prme. Clearly V has proper row nterval matrx deals. However (0) s a trval row nterval matrx deal of V. If V has no proper row nterval matrx deal then we call V to be a deally smple row nterval matrx semgroup. We have an nfnte class of nterval matrx semgroups whch are not deally smple row nterval matrx semgroups. THEOREM 2.4: Let V = {([0, a 1 ], [0, a 2 ],, [0, a n ]) / a Z p ; p a prme; 1 n} be a 1 n row nterval matrx semgroup. V s not an dealy smple 1 n row nterval matrx semgroup. Snce we need p to be prme, we have nfnte number of nterval row matrx semgroups whch are not deally smple (number of prmes s nfnte). How ever we gve some more examples of these and subsemgroups n them under addton. 20

22 Example 2.37: Let V = 1] ] 10 ] a Z ; be a 10 1 column nterval matrx semgroup. Clearly V s of fnte order for V has only fnte number of elements n them. Example 2.38: Let V = 1] 2 ] 3 ] 4 ] 5 ] 6 ] a Z {0};16 be a 6 1 column nterval matrx semgroup under addton. Clearly V s of nfnte order. Example 2.39: Let V = 1] ] 1] 1 a1 Z7 1] be a 4 1 column nterval matrx semgroup under addton. V s of order seven. We have seen examples of these semgroups. The noton of subsemgroups can be defned as n case of row matrx nterval 21

23 semgroups. So we leave ths smple task to the reader but gve examples of these substructures. Example 2.40: Let V = 1] 2] az 30; ] be a 12 1 column nterval matrx semgroup. Take P = 1] ] 12 ] 2 a {0, 2, 4, 6,8,10,12,..., 28} Z30 V; P s a 12 1 column nterval matrx subsemgroup of V. Example 2.41: Let V = 1] 2] a Z {0}; ] be a 12 1 column nterval matrx semgroup. W = 1] ] 12 ] a 7Z {0};1 12 V; 2 22

24 W s a 12 1 column nterval matrx subsemgroup of V. Example 2.42: Let V = 1] ] 7 ] a Z ; be a 7 1 column nterval matrx semgroup. Take 1] 2] I = a {0,2,4,...,34} Z 36;1 7 7 ] V; I s a 7 1 column nterval matrx subsemgroup of V. Example 2.43: Let V = 1] ] 9 ] a Z {0};1 9 2 be a 9 1 column matrx nterval semgroup. Take I = 1] ] 9] a 7Z {0};1 9 V, 2 23

25 I s a 9 1 column nterval matrx subsemgroup of V of nfnte order. Example 2.44: Let V = 1] ] 11] a Z ; be a 11 1 column matrx nterval semgroup. I = 1] ] 11] a {0, 3, 6, 9} Z ; V; s a 11 1 column matrx nterval subsemgroup of V. Example 2.45: Let 1] 2 ] V = a Z 3;18. 8 ] V s a 8 1 column matrx nterval semgroup. Take ] ] W = a Z3 V; ] 24

26 W s a 8 1 column matrx nterval subsemgroup of V. We can defne m n matrx nterval semgroup. Let V = {M = (m j ) m j = [0, a j ]; 1 n and 1 j m, a j Z + {0} (or Z n or R + {0} or Q + {0}} be a collecton of n m nterval matrces. Defne on V matrx addton.e., f M = ([0, a j ]) and N = ([0, b j ]) then M + N = ([0, a j + b j ]) V under nterval matrx addton s a semgroup called the n m matrx nterval semgroup. If m = n then V can be a semgroup under multplcaton as well as addton. We wll descrbe both the operaton wth some nterval matrces. Let [0,5] [0,1] [0,3] [0,2] [0,4] [0,7] A = [0,1] [0,6] [0,5] [0,0] [0,2] [0,8] and [0,1] [0,2] [0,3] [0,4] [0,5] [0,6] B = [0,7] [0,8] [0,1] [0,2] [0,4] [0,5] be nterval matrces wth entres from Z 9. Now A + B = [0,5] [0,1] [0,3] [0,2] [0,4] [0,7] [0,1] [0,6] [0,5] [0,0] [0,2] [0,8] + [0,1] [0,2] [0,3] [0,4] [0,5] [0,6] [0,7] [0,8] [0,1] [0,2] [0,4] [0,5] = [0,6] [0,3] [0,6] [0,6] [0,0] [0,4]. [0,8] [0,5] [0,6] [0,2] [0,6] [0,4] 25

27 Clearly the product s not defned. Now [0, a] [0, b] = [0, ab]. If we take [0,5] [0,7] A = [0,1] [0, 4] and [0,3] [0,1] B = [0,5] [0,8] wth entres from Z + {0}, then AB = [0,5] [0,7] [0,1] [0,4] [0,3] [0,1] [0,5] [0,8] = [0,5][0,3] [0, 7][0,5] [0,5][0,1] [0, 7][0,8] [0,1][0,3] [0, 4][0,5] [0,1][0,1] [0, 4][0,8] = [0,15] [0,35] [0,5] [0,56] [0,3] [0, 20] [0,1] [0,32] = [0,50] [0,61] [0, 23] [0,33]. Thus nterval matrx addton and multplcaton are well defned. Now we wll examples of these structures. Example 2.46: Let 1] 2] 3] 4] V = a Z 20;18 5] 6] 7] 8] 26

28 be a 4 2 nterval matrx semgroup under nterval matrx addton. Example 2.47: Let 1] 4] 7] 10] V [0, a 2] [0, a 5] [0, a 8] [0, a 11] az {0};1 12 3] 6] 9] 12] be a 3 4 nterval matrx semgroup under addton. Example 2.48: Let V = 1] 2] 3] 4] 5] 6] a Z 30;1 9 7] 8] 9] be a 3 3 nterval matrx semgroup under multplcaton. Thus we can as n case of other nterval semgroups defne nterval matrx subsemgroups and deals. Ths task of defnng and gvng examples s left as an exercse for the reader. Now havng seen nterval matrx semgroups we now put forth some of the mportant propertes about these semgroups. An nterval matrx semgroup V s sad to be an nterval matrx Smarandache semgroup (nterval matrx S-semgroup) or Smarandache matrx nterval semgroup (S-matrx nterval semgroup) f V has a proper subset P where P s a group under the operatons of V. We say V s a Smarandache commutatve matrx nterval semgroup f every proper subset A of V whch s a group under the operatons of V s a commutatve matrx nterval group. If only one subset A of V s a group and s commutatve we call V to be a weakly commutatve matrx nterval S-semgroup. 27

29 Example 2.49: Let V = 1] 2] 3] 4] a Z 12;14 be the matrx nterval semgroup under matrx multplcaton. Clearly V has atleast one subset ] ] P = a Z12 ] ] V; P s a matrx nterval commutatve subsemgroup of V; hence P s a weakly commutatve matrx nterval semgroup. Example 2.50: Let V = {([0, a 1 ], [0, a 2 ],, [0, a n ]) a Z + {0}; 1 n} be a row matrx nterval semgroup. Clearly V s a row matrx nterval commutatve semgroup. Example 2.51: Let V = {([0, a 1 ], [0, a 2 ],, [0, a n ]) / a Z 7 \ {0}; 1 n} be a row matrx nterval semgroup. Take W = {([0, a], [0, a],, [0, a]) a Z 7 \ {0}} V. W s a row matrx nterval group of V under the operatons of V. Hence V s a Smarandache row matrx nterval semgroup. THEOREM 2.5: Let V = {([0, a 1 ], [0, a 2 ],, [0, a n ]) / a Z p \ {0}; p s a prme; 1 n} be a row matrx nterval semgroup. Take W = {([0, a], [0, a],, [0, a]) a Z p \ {0}} V; W s a row matrx nterval group. Hence V s a row matrx nterval Smarandache semgroup. The proof s left as an exercse to the reader. THEOREM 2.6: Let V = {([0, a 1 ], [0, a 2 ],, [0, a n ]) / a Z m } m a composte number be a row matrx nterval semgroup. If Z m s a Smarandache semgroup wth P Z m ; P a group then W = {([0, a], [0, a],, [0, a]) a P Z m } V; W s a nterval 28

30 group. Thus V s a row matrx nterval Smarandache semgroup. Ths proof s also left as an exercse for the reader. Example 2.52: Let Z 30 = {0, 1, 2,, 29} be a semgroup under multplcaton modulo 30. V = {([0, a 1 ], [0, a 2 ],, [0, a 9 ]) a Z 30 ; 1 9} be a row nterval matrx semgroup. W = {([0, a 1 ], [0, a 2 ],, [0, a 9 ]) a {0, 5, 10, 15, 20, 25} Z 30 } V. W s a row nterval matrx deal of V. THEOREM 2.7: Let V = {([0, a 1 ], [0, a 2 ],, [0, a n ]) / a Z p } be a row nterval matrx semgroup under multplcaton; V has no proper deals. Ths proof s also left for the reader. Let V = 1] n] [0,b ] [0,b ] [0,c 1] [0,c n] a,b,c Z;1 n 1 n n be the collecton of all n n nterval square matrx. V s a square matrx nterval semgroup under multplcaton. (or addton, or used n the mutually exclusve sense). Example 2.53: Let 1] 2] V = 3] 4] a Z 4;14 be a 4 4 nterval matrx semgroup under addton modulo 4. (V s also a 4 4 nterval matrx semgroup under multplcaton). For take 29

31 and n V. [0,3] [0,1] A = [0, 2] [0, 2] [0,1] [0, 2] B = [0, 2] [0,3] A B = [0,3] [0,1] [0, 2] [0, 2] [0,1] [0, 2] [0, 2] [0,3] = [0,3][0,1] [0,1][0, 2] [0,3][0, 2] [0,1][0,3] [0, 2][0,1] [0, 2][0, 2] [0, 2][0, 2] [0, 2][0,3] = [0,3] [0, 2] [0, 2] [0,3] [0, 2] [0,0] [0,0] [0, 2] = [0,1] [0,1]. [0, 2] [0, 2] We can defne the noton of Smarandache Lagrange semgroup, Smarandache subsemgroup, Smarandache hypersubsemgroup Smarandache p-sylow subgroup, Smarandache Cauchy elements of a S-semgroup and Smarandache coset n case of nterval matrx semgroup n an analogous way [9]. We wll llustrate these stuatons by examples for the defnton s very smlar to that of semgroups [9]. Example 2.54: Let V = 1] 2] 3] 4] 1] 6] a 0; 2,3,4,6,7,8 7] 8] 1] 30

32 be a 3 3 nterval matrx semgroup under multplcaton. ] ] ] W A ] ] ] A 0;a {1,3} Z ; a 0f 1 7] 8] 9] V s a subgroup. Clearly V s a Smarandache matrx nterval semgroup. Example 2.55: Let V = 1] 2] 3] 4] a Z 9;14 be a square matrx nterval semgroup. Take W = 1] 2] a 1{1,8},a 0Z 9; 3] 1] A 0;23 V; V s a square matrx nterval group under multplcaton. Thus V s a Smarandache square matrx nterval semgroup. Now we proceed onto gve examples of Smarandache matrx nterval subsemgroup or matrx nterval Smarandache semgroup. Example 2.56: Let V = [0, a 1] [0, a 2] [0, a 3] [0, a 4] ] ] ] ] 13 ] 14 ] 15 ] 16 ] a Z ; ] 10] 11] 12] 31

33 be a 4 4 matrx nterval semgroup. Take 1] 2] 3] 4] A 0;a 0; ] ] ] ] a {1,...,10} Z W 9] 10] 1] 12] 2,3,4,5,7,8,9, 13] 14] 15] 1] 10,12,13,14, V; W s a nterval matrx group. Now take P = {All 4 4 square matrces wth ntervals of the form [0, a ]; where a Z 11 \ {0}} V; P s a nterval matrx subsemgroup of V and W P; so P s a matrx nterval Smarandache subsemgroup of V. Example 2.57: Let V = 1] 2] 3] 4] a Z 15;14 be a nterval matrx semgroup. Take W = [0, a 1] [0, a 2] a 1{0,3, 6,9,12} Z 15; 3] 4] a 0;24 V to be a nterval matrx subsemgroup of V. Let P A [0, a 1] [0, a 2] a 1{0,3, 6,9,12} Z 15; 3] 1] A 0;a 2 a3 0 W; W s a nterval matrx smarandache subsemgroup of V. 32

34 Now we proceed onto gve examples of the noton of Smarandache nterval matrx subsemgroup. Example 2.58: Let V = {set all 5 5 nterval matrces wth ntervals of the form [0, a ] wth a Z 43 } be a nterval matrx semgroup. Take W = {A / all 5 5 nterval matrces wth ntervals of the form [0, a 1 ] wth a 1 Z 43 \ {0} such that A 0. A s a 5 5 dagonal nterval matrx} V; W s a group under nterval matrx multplcaton. So V s a nterval matrx Smarandache semgroup. Further f we take P = {all 5 5 dagonal nterval matrces wth ntervals of the form [0, a ] wth a Z 43 \ {0}} V then P s a nterval matrx subsemgroup of V. We see W P and W s the largest nterval matrx group present n P. Thus P s a matrx nterval Smarandache subsemgroup of V. Example 2.59: Let V = 1] 2] 3] 4] az 11;14 be a matrx nterval semgroup. Let P = 1] 2] 3] 1] a1z 11; A 0; a2 a3 0 V; P s the largest nterval matrx group present n V; but V has no proper matrx nterval subsemgroup whch contans P. Example 2.60: Let V = {[0, a ] a Z 43 } be a matrx nterval semgroup. P = {[0, a ] a Z 43 \ {0}} s the matrx nterval 33

35 subgroup of V. Infact V has no proper matrx nterval subsemgroup contanng P. Example 2.61: Let V = {[0, a ] / a Z 6 } be the matrx nterval semgroup. Take W = {[0, 1], [0, 5]} V s a nterval subgroup of V. Take P = {[0, 1], [0, 5], [0, 0]} V s a nterval Smarandache subsemgroup we see P s a nterval Smarandache hyper subsemgroup of V. It s left for the reader to prove the followng theorems. THEOREM 2.8: Let V = {all n n dagonal nterval matrces wth ntervals of the form [0, a 1 ], a 1 Z p ; all dagonal elements are the same} s a Smarandache smple nterval matrx semgroup whch s a Smarandache nterval matrx semgroup. THEOREM 2.9: Let V be a Smarandache matrx nterval semgroup. Every Smarandache matrx nterval hyper subsemgroup s a Smarandache matrx nterval subsemgroup but every Smarandache matrx nterval subsemgroup n general s not a S-matrx nterval hyper subsemgroup. Now we proceed onto gve examples of Smarandache matrx nterval Lagrange semgroup (S-matrx nterval Lagrange semgroup). Example 2.62: Let V = {[0, a] a Z 4 } be a S nterval semgroup. A = {[0, 1], [0, 3]} V s a nterval subgroup of V. Clearly o(a) / o(v) so V s a S-matrx nterval Lagrange semgroup. Example 2.63: Let V = {[0, a] a Z 9 } be a S-matrx nterval semgroup. Let A = {[0, 1], [1, 8]} V and B = {[0, 1], [0, 2], [0, 4], [0, 5], [0, 7], [0, 8]} V be subgroup of V. We see both of them do not dvde of the order of V. So V s not a S-nterval Lagrange semgroup. 34

36 Example 2.64: Let V = {[0, a] a Z 10 } be S-nterval semgroup. V s a S-weakly Lagrange nterval semgroup. The proof of the followng theorem s left as an exercse for the reader. THEOREM 2.10: Every S-nterval Lagrange semgroup s a S- nterval weakly Lagrange semgroup. Next we proceed onto llustrate S-p-Sylow nterval subgroup of a S-nterval semgroup. Example 2.65: Let V = {[0, a] a Z 16 } be a S-nterval semgroup. A = {[0, 1], [0, 9]} V s a nterval subgroup of V. 2/ o(v) but 2 2 / o(v), but V has S-2-Sylow nterval subgroups of order 4 gven by B = {[0, 6], [0, 2], [0, 4], [0, 8]} V; 4 / o(v). We see n case of S-nterval semgroup V we say f p s a prme such that p / o(v) then we can have nterval subgroup of order p ; where p / o(v), we call such ntervals subgroups of the S-nterval semgroup to be S-p-Sylow nterval subgroups of V. We gve examples of S-Cauchy element of a nterval semgroup. We see a S-Cauchy element of a nterval semgroup x of V s such that x t = 1 and t / o(v). Example 2.66: Let V = {[0, a] a Z 19 } be a S-nterval semgroup. Take x = [0, 18] V; x 2 = ([0, 18]) 2 = [0, 1]. We see 2 / o(v). Thus x s not a S-Cauchy nterval element of V. We leave the proof of the followng theorem of the reader. 35

37 THEOREM 2.11: Let V = {[0, a] / a Z p }; (p a prme) be the S- nterval semgroup under multplcaton. No element of V s a S- Cauchy element of V. The proof s obvous from the fact that no nteger n can dvde the prme p. Hence the clam. 36

38 Chapter Three INTERVAL POLYNOMIAL SEMIGROUPS In ths chapter we ntroduce the noton of nterval polynomal semgroups. We call a polynomal n the varable x to be an nterval polynomal f the coeffcents of x are ntervals of the form [0, a ] / a Z p (or Z n or Z + {0} or Q + {0} or R + {0}. [0, 5] + [0, 7]x + [0, 2] x 3 + [0, 14] x 9 = p (x) s a nterval polynomal n the varable x. We now defne nterval polynomal semgroup under addton (or multplcaton). We just llustrate how nterval polynomals are added. Let p (x) = [0, 2] + [0, 3] x 2 + [0, 7] x 7 + [0, 11] x 9 and q (x) = [0, 12] + [0, 7] x + [0, 14] x 3 + [0, 10] x 7 + [0, 5] x 8 + [0, 12] x 9 + [0, 5] x

39 p (x) + q (x) = ([0, 2] + [0, 3] x 2 + [0, 7] x 7 + [0, 11] x 9 + [0, 12] + [0, 7] x + [0, 14] x 3 + [0, 10] x 7 + [0, 5] x 8 + [0, 12] x 9 + [0, 5] x 20 ) = ([0, 2] + [0, 12]) + [0, 7] x + [0, 3] x 2 + [0, 14]x 3 + ([0, 7] x 7 + [0, 10] x 7 ) + [0, 5]x 8 + ([0, 11] x 9 + [0, 12] x 9 ) + [0, 5] x 20 = [0, 14] + [0, 7]x + [0, 3] x 2 + [0, 14] x 3 + [0, 17] x 7 + [0, 5] x 8 + [0, 23] x 9 + [0, 5] x 20. Now we wll just defne nterval polynomal multplcaton. p (x) = [0, 3] + [0, 5] x 2 + [0, 11] x 5 and q (x) = [0, 8] + [0, 1] x + [0, 9] x 3. p(x).q(x) = ([0, 3] + [0, 5] x 2 + [0, 11] x 5 ) ([0, 8] + [0, 1]x + [0, 9] x 3 ). = [0, 3] [0, 8] + [0, 5] [0, 8] x 2 + [0, 11] [0, 8] x 5 + [0, 3] [0, 1] x + [0, 5] x 2 [0, 1] x + [0, 11] x 5 [0, 1] x + [0, 3] [0, 9] x 3 + [0, 5] x 2 [0, 9] x 3 + [0, 11] x 5 [0, 9] x 3. = [0, 24] + [0, 40] x 2 + [0, 88] x 5 + [0, 3] x + [0, 5] x 3 + [0, 11] x 6 + [0, 27]x 3 + [0, 45] x 5 + [0, 99] x 8. = [0, 24] + [0, 3]x + [0, 40]x 2 + [0, 32]x 3 + [0, 45]x 5 + [0, 11]x 6 + [0, 99] x 8. Now havng defned nterval polynomal addton and multplcaton we proceed onto defne nterval polynomal semgroup under these operatons. DEFINITION 3.1: Let S = n 0 ]x a Z (or Z + {0} n or R + {0}, or Q + {0}, C + {0}) and x s a varable or 38

40 ndetermnate} S under addton of nterval polynomals s a semgroup defned as nterval polynomal semgroup. We wll llustrate ths stuaton by some examples. Example 3.1: Let S = 9 ]x a Z {0} 0 be a nterval polynomal semgroup under addton. Clearly the number of elements n S s nfnte so S s an nfnte order nterval polynomal semgroup. Example 3.2: Let S = 3 0 ]x a Z 11 be a nterval polynomal semgroup. Clearly S s of fnte order. We see clearly the nterval polynomal semgroups gven n the above examples are not compatble under multplcaton. Example 3.3: Let S = 7 ]x a R {0} 0 be the nterval polynomal semgroup. S s an nfnte nterval polynomal semgroup under addton. Now we can defne substructures for these structures. DEFINITION 3.2: Let S be a nterval polynomal semgroup under addton. Suppose W S be a proper subset of S and f W 39

41 s tself an nterval polynomal semgroup under addton then we defne W to be an nterval polynomal subsemgroup of S. We wll llustrate ths stuaton also by some examples. Example 3.4: Let S = 8 ]x a Z {0} 0 be a nterval polynomal semgroup under addton. Take W = 8 ]x a 3Z {0} S; 0 W s a nterval polynomal subsemgroup of S under addton. Example 3.5: Let S = 20 0 ]x a Z 12 be a nterval polynomal semgroup under addton. Take W = 10 ]x a Z12 S; 0 W s a nterval polynomal subsemgroup of S under addton. Both S and W are of fnte order. We can defne deals as n case of usual semgroups. If a nterval polynomal semgroup S has no proper nterval polynomal subsemgroups we call S to be a smple nterval polynomal semgroup. 40

42 We now proceed onto defne polynomal nterval semgroup under multplcaton. DEFINITION 3.3: Let V = ]x az {0} 0 (or Z n or R + {0}, or Q + {0}, x a varable} be a collecton of nterval polynomals. If product s defned on V then V s a nterval polynomal semgroup under multplcaton. We wll llustrate ths stuaton by some examples. Example 3.6: Let S = 0 ]x a Z8 be a polynomal nterval semgroup under multplcaton. Clearly S s of nfnte order. Example 3.7: Let S = ]x a R {0} 0 be a nterval polynomal semgroup under multplcaton. Clearly S s of nfnte order. Substructure s defned as n case of usual semgroups. However we wll llustrate ths stuaton by some examples. Example 3.8: Let S = ]x a Q 0 41

43 be a nterval polynomal semgroup. Take P = ]x a Z S; 0 P s a nterval polynomal subsemgroup of S. Clearly P s a not as nterval polynomal deal of S. Example 3.9: Let S = 0 ]x a Z 30 be a nterval polynomal semgroup under multplcaton. Take T = [0, a ]x a {0, 2, 4, 6,..., 26, 28} Z30 S; 0 T s a nterval polynomal subsemgroup of S. T s also a nterval polynomal deal of S. Thus we can have nterval polynomal subsemgroups whch are not nterval deals of the polynomal semgroup. Example 3.10: Let S = ]x a Q {0} S; 0 T s only a polynomal nterval subsemgroup and s not a polynomal nterval deal of S. Infact S has no polynomal nterval deals but has nfnte number of polynomal nterval subsemgroups. Now havng seen the two substructures we proceed on to consder fnte polynomal nterval semgroups whch are Smarandache Lagrange polynomal nterval semgroup, Smarandache polynomal nterval semgroup and so on. It s pertnent to menton here that polynomal nterval semgroup can contan Smarandache Cauchy elements. 42

44 We wll llustrate ths by some examples. Example 3.11: Let S = 2 0 ]x a Z3 be a polynomal nterval semgroup under addton. S = {0, [0, 1] x, [0, 1] x 2, [0, 2] x, [0, 2] x 2, [0, 1], [0, 2], [0, 1] + [0, 1]x, [0, 1] + [0, 1] x 2, [0, 1] + [0, 2] x, [0, 1] + [0, 2] x 2, [0, 2] + [0, 1] x, [0, 2] + [0, 2] x 2 [0, 2] + [0, 1]x 2, [0, 2] + [0, 2]x, [0, 1] + [0, 1]x + [0, 1] x 2, }. We see [0, 1] + [0, 1] + [0, 1] = 0 [0, 1] x + [0, 1]x + [0, 1]x = 0 [0, 2]x + [0, 2]x + [0, 2]x = 0 Thus we have several Cauchy elements, S s also a S- polynomal nterval semgroup. It s left as an exercse for the reader to fnd the order of S and fnd out whether the elements are S-Cauchy elements of S. Example 3.12: Let S = 5 0 ]x a Z2 be a polynomal nterval semgroup under addton. We see S has several elements of fnte order but one s to fnd the order of S. S s a S-polynomal nterval semgroup. We see [0, 1] + [0, 1] = 0, ([0, 1]x + [0, 1]) + ([0, 1]x + [0, 1]) = 0 and so on. The reader s left wth the task of fndng the order of S. However S s a S-nterval polynomal semgroup. 43

45 Example 3.13: Let S = 0 ]x a Z7 be a polynomal nterval semgroup. We see S s of nfnte order (S be under addton or multplcaton). We cannot n ths case defne S Cauchy element. However S s a S-polynomal nterval semgroup under addton and S s a S-polynomal nterval semgroup under multplcaton. Example 3.14: Let S = 8 0 ]x a Z8 be a nterval polynomal semgroup under addton. S s a S- nterval polynomal semgroup. S s a S-commutatve nterval polynomal semgroup. Further t s easly verfed S s a S-weakly nterval polynomal semgroup. For [0, 1]x n S generates a cyclc group under addton where 1 8. Example 3.15: Let S = 6 7 ]x x 1, x 8 = x, so on; a Z 6 } 0 be a polynomal nterval semgroup under multplcaton. For f p(x) = [0, 1]x + [0, 5]x 5 + [0, 2] and q(x) = [0, 4] + [0, 3]x 3 n S the p(x)q(x) = ([0, 2] + [0, 1]x + [0, 5]x 5 ) ([0, 4] + [0, 3]x 3 ) = [0, 2] [0, 4] + [0, 2] [0, 3]x 3 + [0, 1]x [0, 4] + [0, 1]x [0, 3]x 3 + [0, 5]x 5 [0, 4] + [0, 5]x 5. [0, 3]x 3 = [0,2] + [0,0] x 3 + [0,4]x + [0,3]x 4 + [0,2] x 5 + [0,3]x. = [0,2] + [0,1]x + [0,3]x 4 + [0,2]x 5. 44

46 It s easly verfed S s a S-nterval polynomal weakly cyclc semgroup. Ths smple result can be proved by the reader. Example 3.16: Let S = 0 ]x a Z 12 be nterval polynomal semgroup under multplcaton. S has nterval polynomal deals, for take P = 0 [0, a ]x a {0, 2, 4, 6,8,10} Z 12 S s an nterval polynomal deal of S. Example 3.17: Let S = 0 ]x a Z 12 be a nterval polynomal semgroup under addton. Clearly S has only nterval polynomal subsemgroups and has no deals. P = 0 [0, a ]x a {0, 2, 4, 6,8,10} Z 12 S s not an nterval polynomal deal of S. Example 3.18: Let S = ]x a Z 12; x x 1, x 4 = x and so on} 0 be a polynomal nterval semgroup. Fnd order of S. Is S a S- Lagrange nterval polynomal semgroup? 45

47 Example 3.19: Let S = ]x a Z 2; x x 1, x 5 = x} 0 be polynomal nterval semgroup under multplcaton. Clearly S = {[0, 1]x, 0, [0, 1], [0, 1]x 2, [0, 1]x 3, [0, 1] + [0, 1]x [0, 1] + [0, 1]x 2, [0, 1] + [0, 1]x 3, [0, 1]x + [0, 1]x 2, [0, 1]x + [0, 1]x 3, [0, 1]x 2 + [0, 1] + [0, 1]x + [0, 1]x 3, [0, 1] + [0, 1]x 2 + [0, 1]x 3, [0, 1]x + [0, 1]x 2 + [0, 1]x 3, [0, 1] + [0, 1]x + [0, 1]x 2 + [0, 1]x 3 }, and o (S) = 16. T = {[0, 1]x, [0, 1]x 2, [0, 1]x 3, [0, 1]} S s a nterval polynomal subgroup of S. P = {[0, 1]x 2, [0, 1]} S s also a nterval polynomal subgroup of S. Thus S s a S-nterval polynomal semgroup. Infact S s a commutatve nterval polynomal semgroup wth dentty [0, 1]. Further I = {0, [0, 1] + [0, 1]x + [0, 1]x 2 + [0, 1]x 3 } S s a nterval polynomal deal of S. Now havng seen examples of polynomal nterval semgroups we now proceed onto defne symmetrc nterval semgroups or permutaton nterval semgroup or nterval permutatve semgroup n the followng chapter. 46

48 Chapter Four SPECIAL INTERVAL SYMMETRIC SEMIGROUPS In ths chapter we for the frst tme ntroduce the noton of mappng of n row ntervals ([0, a 1 ],, [0, a n ]) to tself. Ths forms the semgroup under the composton of mappngs and s somorphc wth the symmetrc semgroup S(n). We also defne specal nterval symmetrc group and study some propertes related wth them. DEFINITION 4.1: Let X = {[0, a 1 ], [0, a 2 ],, [0, a n ]} be a set of n dstnct ntervals. We say : X X s an nterval mappng f ([0, a ]) = [0, a j ]; 1, j n. Let S (X) denote the collecton of all nterval mappngs of X to X. S (X) under the composton of nterval mappngs s a semgroup defned as the nterval symmetrc semgroup. We wll frst llustrate ths stuaton by some examples. 47

49 Example 4.1: Let X = {[0, a 1 ], ]0, a 2 ]} be the nterval set a 1 a 2. The set of all maps of X to X are as follows: 1 : X X gven by 1 ([0, a 1 ]) = [0, a 1 ] and 1 ([0, a 2 ]) = [0, a 2 ]. 2 : X X s gven by 2 ([0, a 1 ]) = [0, a 2 ] and 2 ([0, a 2 ]) = [0, a 1 ], 3 : X X s defned by 3 ([0, a 1 ]) = [0, a 1 ] and 3 ([0, a 2 ]) = [0, a 1 ]. 4 : X X s such that 4 ([0, a 1 ]) = [0, a 2 ] and 4 ([0, a 2 ]) = [0, a 2 ]. Thus S (X) = { 1, 2, 3, 4 }; and S (X) under composton of maps s an nterval symmetrc semgroup. Clearly S (X) = 2 2 = 4. Example 4.2: Let X = {[0, a 1 ], [0, a 2 ], [0, a 3 ]}; a a j f j a > 0; 1 3. The maps of X to X s S(X) = { 1, 2, 3, 4, 5, 6, 7,, 26, 27 }. 1 ([0, a ]) = [0, a ]; = 1, 2, 3; 2 ([0, a 1 ]) = [0, a 2 ], 2 ([0, a 2 ]) = [0, a 3 ], 2 ([0, a 3 ]) = [0, a 1 ], 3 ([0, a 1 ]) = [0, a 1 ]; 3 ([0, a 2 ]) = [0, a 3 ], 3 ([0, a 3 ]) = [0, a 2 ]; 4 ([0, a 1 ]) = [0, a 2 ], 4 ([0, a 2 ]) = [0, a 1 ]; 4 ([0, a 3 ]) = [0, a 3 ], 5 ([0, a 1 ]) = [0, a 3 ]; 5 ([0, a 2 ]) = [0, a 2 ], 5 ([0, a 3 ]) = [0, a 1 ]; 6 ([0, a 1 ]) = [0, a 3 ], 6 ([0, a 2 ]) = [0, a 1 ]; 6 ([0, a 3 ]) = [0, a 2 ], 7 ([0, a 1 ]) = [0, a 1 ]; 7 ([0, a 2 ]) = [0, a 1 ], 7 ([0, a 3 ]) = [0, a 1 ],, 27 ([0, a 3 ]) = [0, a 3 ] 27 ([0, a 1 ]) = [0, a 3 ] and 27 ([0, a 2 ]) = [0, a 3 ]. Thus o(s (X)) = 27 = 3 3. We see S (X) s a nterval symmetrc group of mappngs of the nterval set X to tself. We can n general say f X = {[0, b 1 ], [0, b 2 ],, [0, b n ]} wth b b j ; j, (1, j n) then S(X) s the nterval symmetrc semgroup of order n n. 48

50 We have the followng nterestng theorem and observatons when we say the nterval [0, a] s mapped on to [0, b] we mean the contnuous nterval segment 0 to a s mapped onto the contnuous nterval segment 0 to b. We see the map may contract or extend the nterval for nstance [0, 5] s mapped to [0, 2 ] then certanly a contracton has taken place or we can realze the map s not an embeddng On the other hand f [0, 2 ] nterval s mapped onto [0, 5] we can realze t as expanson. All these maps wll be useful when we use the concept of fnte element methods lke, stffness matrces or n any other applcatons. However we see we have an somorphsm between S (n) and S(X) where X = {[0, a 1 ], [0, a 2 ],, [0, a n ]} all ntervals are dstnct a > 0 and a a j, f j; 1, j n. After all : [0, a t ] [0, a p ] are only maps. 1 t, p n. Keepng ths n mnd we have the followng. THEOREM 4.1: Let S (n) be the symmetrc semgroup on the set {1, 2,, n} and S (X) be the nterval symmetrc semgroup on the set X = {[0, a 1 ], [0, a 2 ],, [0, a n ]}, a a j, j, a > 0, 1, j n. Then S (n) s somorphc wth S (X). Proof : We know the symmetrc semgroup S(n) s of order n n. Now the order of the nterval symmetrc semgroup S(X) where X = {[0, a 1 ],, [0, a n ]} s also of order n n. Now f we put [0, a ] = x, = 1, 2,, n and S(n) s the set of all maps of (1, 2,, n) to tself and assocate each to x ; 1 n, we see the one to one correspondence between the maps. Thus S(n) S(X). We wll llustrate ths n case of S(2). 49

51 Example 4.3: Let S(X) = { 1, 2, 3, 4 } where : {1, 2} {1, 2}; = 1, 2, 3, 4. 1 (1) = 1, 1 (2) = 2, 2 (1) = 1, 2 (2) = 1, 3 (1) = 2, 3 (2) = 2, 4 (1) = 2 and 4 (2) = 1 s the symmetrc semgroup of order 2 2 = 4. Now X = {[0, a 1 ], [0, a 2 ]} be the nterval set a 1 a 2 and a > 0; = 1, 2. S(X) = 1, 2, 3, 4 } where I : X X; = 1, 2, 3, 4. 1 ([0, a 1 ]) = [0, a 1 ], 1 = ([0, a 2 ]) = [0, a 2 ] 2 ([0, a 1 ]) = [0, a 1 ], 2 = ([0, a 2 ]) = [0, a 1 ] 3 ([0, a 1 ]) = [0, a 2 ], 3 = ([0, a 2 ]) = [0, a 2 ] and 4 ([0, a 1 ]) = [0, a 2 ], 4 = ([0, a 2 ]) = [0, a 1 ]. S(X) s the nterval symmetrc semgroup of order 2 2 = 4. Now defne a map : S (2) S (X) as follows. ( 1 ) = 1, ( 2 ) = 2 ( 3 ) = 3 and ( 4 ) = 4. It s easly verfed s a semgroup homomorphsm, nfact an somorphsm. Hence the clam. We wll enumerate some of the propertes enjoyed by the nterval symmetrc semgroup. Example 4.4: Let S(X) be the set of all maps from the three element nterval set X = {[0, a 1 ], [0, a 2 ], [0, a 3 ]} to tself. Clearly S (X) s the semgroup under the operaton of composton of map. Thus S (X) s the symmetrc nterval semgroup of order 3 3 = 27. We see S (X) s S-symmetrc nterval semgroup as t has 5 nterval subgroups. For take P 1 = { 1, 2 } where 1 : 1] 1] 2] 2] 3] 3] and 50

52 2 : 1] 1] 2] 3] 3] 2] P 2 = { 1, 3 } where s gven by and 3 : 1] 3] 2] 2] 3] 1] P 3 = { 1, 4 } where 1 s gven above 4 : 1] 2] 2] 1] 3] 3] P 4 = { 1, 5, 6 } where 1 s the dentty map and and 5 : 6 : 1] 2] 2] 3] 3] 1] 1] 3] 2] 1] 3] 2] and P 5 = { 1, 2, 3, 4, 5, 6 }. Thus S(X) has P 1, P 2, P 3, P 4 and P 5 to be 5 nterval subgroups of whch P 1, P 2, P 3 and P 4 are cyclc. Thus S (X) s only a S-nterval symmetrc weakly cyclc semgroup as P 5 s not an abelan group. In vew of ths we have the followng theorems the proof of whch are left as exercses for the reader. 51

53 THEOREM 4.2: Let S (X) be a nterval symmetrc semgroup where X = {[0, a 1 ], [0, a 2 ],, [0, a n ]}, a a j ; j, 1<, j<n. S(X) s a S-weakly cyclc nterval symmetrc semgroup. THEOREM 4.3: Let S (X) be a nterval symmetrc semgroup where X = {[0, a 1 ], [0, a 2 ],, [0, a n ]}, a a j, j, 1<, j<n. S(X) s a S-nterval symmetrc semgroup. THEOREM 4.4: Let S (X) be a nterval symmetrc semgroup where X = {[0, a 1 ], [0, a 2 ],, [0, a n ]}, a a j f j, a > 0, 1, j n. S(X) s only a Smarandache weakly commutatve nterval symmetrc semgroup. Proof: Take n S (X), P the collecton of all one to one mappng of X to tself, then P s a nterval symmetrc subgroup of S (X) but s not a commutatve nterval symmetrc group. Hence S (X) s only a S-weakly commutatve nterval symmetrc semgroup. We frst proceed onto gve the basc defnton of nterval symmetrc group. DEFINITION 4.2: Let X = {[0, a 1 ], [0, a 2 ],, [0, a n ]} be an nterval set S X denote the set of all one to one maps of the nterval set X. S X under the composton of mappngs s a group, whch wll be known as the nterval symmetrc group. Example 4.5: Let S X = { 1, 2, 3,, 6 } where X = {[0, a 1 ], [0, a 2 ], [0, a 3 ]} s the nterval set. S X s the nterval symmetrc group wth 1 ([0, a ]) = [0, a ]; ([0, a 1 ]) = [0, a 1 ], 2 ([0, a 2 ]) = [0, a 3 ] and 2 ([0, a 3 ]) = [0, a 2 ]. 3 ([0, a 1 ]) = [0, a 2 ], 3 ([0, a 2 ]) = [0, a 1 ] 3 ([0, a 3 ]) = [0, a 3 ], 4 ([0, a 1 ]) = [0, a 3 ] 4 ([0, a 2 ]) = [0, a 2 ], 4 ([0, a 3 ]) = [0, a 1 ] 5 ([0, a 1 ]) = [0, a 2 ], 5 ([0, a 2 ]) = [0, a 3 ] 5 ([0, a 3 ]) = [0, a 1 ], and 6 ([0, a 1 ]) = [0, a 3 ]; 6 ([0, a 2 ]) = [0, a 1 ] and 6 ([0, a 3 ]) = [0, a 2 ]. 52

54 It s easly verfed S X under the composton of maps s a group, called the nterval symmetrc group. We have the followng theorems whch are left as exercses for the reader to prove. THEOREM 4.5: Let S X be the nterval symmetrc group on X = {[0, a 1 ], [0, a 2 ],, [0, a n ]}, of n dstnct ntervals a > 0, S n S X where S n s the symmetrc group of degree n. THEOREM 4.6: The S nterval symmetrc semgroup S (X) has ts largest nterval group S X to be contaned n the proper nterval subset A = S X { 1, 2,, n } where ([0, a j ]) = [0, a ] for all j = 1, 2,, n. true for =1, 2,, n, whch s an nterval symmetrc subsemgroup of S (X). We wll llustrate ths stuaton by an example. Example 4.6: Let X = {[0, a 1 ], [0, a 2 ], [0, a 3 ], [0, a 4 ]} be a nterval set of cardnalty four. S (X) be the nterval symmetrc semgroup. Consder A = S X 2] 2] 2] 2] 1] 2] 3] 4], 1] 2] 3] 4], 1] 1] 1] 1] 1] 2] 3] 4], 3] 3] 3] 3] 1] 2] 3] 4] 4] 4] 4] 4] 53

55 Clearly A s a proper subset and s the nterval symmetrc subsemgroup of S(X). Further A s a S-hyper nterval symmetrc subsemgroup of S(X). COROLLARY 4.1: S(X) the S-nterval symmetrc semgroup s not a S-smple symmetrc semgroup. We wll be usng the defntons of S-Lagrange semgroup and S-weakly Lagrange semgroup [9]. Example 4.7: Consder the nterval symmetrc semgroup S(X) where X = {[0, a 1 ], [0, a 2 ], [0, a 3 ], [0, a 4 ]} of 4 dstnct ntervals a > 0; = 1, 2, 3, 4. Clearly order of S(X) s 4 4. We see S(X) has S X to be nterval subgroup of order 24. Clearly 24 / 4 4. Thus S(X) s not a S-Lagrange nterval symmetrc semgroup. However S(X) has nterval subgroups of order two and four whch dvdes 4 4. Hence S(X) s a S-nterval symmetrc weakly Lagrange semgroup. THEOREM 4.7: Let S(X) be a nterval symmetrc semgroup on n-dstnct ntervals,.e. X = {[0, a 1 ], [0, a 2 ],, [0, a n ]}, (1) S(X) s a S-nterval symmetrc semgroup. (2) S(X) s not a S-nterval symmetrc Lagrange semgroup. (3) S(X) s a S-nterval symmetrc weakly Lagrange semgroup. (4) S(X) has Smarandache nterval symmetrc p-sylow subgroups provded X has p number of dstnct ntervals and p s a prme. If the number of dstnct ntervals n X s a composte number say n and f p s a prme such that p/n then also S(X) has S-p- Sylow subgroups. THEOREM 4.8: Let S (X) be a nterval symmetrc semgroup. S(X) has S-p-Sylow semgroups. Proof s drect and s left as an exercse to the reader. 54

56 Please refer [9] for Smarandache Cauchy elements of a semgroup. Example 4.8: Let S(X) be a nterval symmetrc semgroup wth X = {[0, a 1 ], [0, a 2 ], [0, a 3 ], [0, a 4 ], [0, a 5 ]} where X has 5- dstnct ntervals, o(s(x)) = 5 5. We have S(X) such that 5 = dentty map and Thus S(X) has Smarandache Cauchy elements. In vew of ths we have the followng results. THEOREM 4.9: Let S(X) be a nterval symmetrc semgroup of order n n where X = {[0, a 1 ], [0, a 2 ],, [0, a n ]}. S(X) has S- Cauchy elements. The proof s left as an exercse for the reader. However we wll llustrate ths stuaton by an example. Example 4.9: Let S(X) be a nterval symmetrc semgroup of order 6 6. S(X) has S-Cauchy elements. For take t = 1] 2] 3] 4] 5] 6] 2] 3] 4] 5] 6] 1] n S(X). Clearly ( t ) 6 = dentty element of S(X). Thus S(X) has S-Cauchy elements. p = 1] 2] 3] 4] 5] 6] 2] 3] 1] 4] 5] 6] n S(X) s such that ( p ) 3 = dentty elements of S(X). Thus S(X) has S-Cauchy elements. In vew of ths we have the followng theorem. 55

57 THEOREM 4.10: Let S(X) be a nterval symmetrc semgroup of order n n ; n a composte number S(X) has S-Cauchy elements. Proof: Take = 1] 2] n1] n] 2] 3] n] 1] n S (X). Clearly ( ) n = dentty element of S (X). Take p n. p a prme then 1] 2] p1] p] p1] n] t 2] 3] p] 1] p1] n] n S(X) s such that ( t ) p = dentty element of S(X). Thus S(X) has S-Cauchy elements. Cayley s theorem for S-semgroups can also be extended n case of S-nterval semgroups. Please refer for S-semgroup homomorphsm and S- semgroup automorphsm [9]. Snce S(X) S(n) where S (n) s a permutaton of (1, 2,, n) and S(X) s a nterval symmetrc semgroup wth n nterval set X = {[0, a 1 ], [0, a 2 ],, [0, a n ]}, a > 0 and a a j f j; 1, j n. We can have Cayley s theorem for S-nterval semgroups. Several nterestng results n classcal group theory can be proved for S-nterval semgroups wth approprate modfcatons. Now we proceed onto defne specal nterval symmetrc semgroups. For now on wards by a specal nterval [a, b] we mean a < b, a 0, a and b postve ntegers. We say X = {[a 1, b 1 ], [a 2, b 2 ],, [a n, b n ]} s a specal nterval set f a < b, a 0; 1 n and each a j < b k for every 1 j, k n; that s all a s are less than b k even f k. We call such nterval collecton to be specal nterval collecton. 56

58 We wll frst llustrate specal ntervals. Example 4.10: Let X = {[7, 12], [5, 10], [3, 8]}, X s a specal nterval set. Suppose X = {[5, 10], [6, 7], [54, 5], [9, 12]}; X s not a specal nterval set as 7 < 9 and 6 < 5 so a < b j s not true for every a and b j. Let X = {[a 1, b 1 ], [a 2, b 2 ]} be the specal nterval set then the nterval set generated by X denoted by X s {[a 1, b 1 ], [a 2, b 2 ], [a 2, b 1 ], [a 2, b 2 ]} s an nterval set. Let X = {[a 1, b 1 ], [a 2, b 2 ], [a 3, b 3 ]} be the specal nterval set, then the nterval set generated by X denoted by X s {{[a 1, b 1 ], [a 2, b 2 ], [a 3, b 3 ], [a 1, b 2 ], [a 1, b 3 ], [a 2, b 1 ], [a 2, b 3 ] [a 3, b 1 ], [a 3, b 2 ]}. Thus we see f X = {[a 1, b 1 ], [a 2, b 2 ],, [a n, b n ]} s a specal nterval set then the nterval set generated by X denoted by X s {[a 1, b 1 ],, [a n, b n ], [a 1, b 2 ], [a 1, b 3 ],, [a 1, b n ],, [a n, b 1 ], [a n, b 2 ],, [a n, b n-1 ]}. Clearly the number of elements n X s n 2. Now we proceed onto defne the noton of specal nterval symmetrc semgroup or nterval specal symmetrc semgroup. DEFINITION 4.3: Let X be a specal nterval set X the nterval set generated by X. S(X) set of all mappngs of X to X. S(X) s defned as the specal nterval symmetrc semgroup or nterval specal symmetrc semgroup. 2 The order of S (X ) = n n 2. We wll llustrate ths stuaton by an example. Example 4.11: Let X = {[a 1, b 1 ], [a 2, b 2 ]} be a specal nterval set wth two dstnct elements. X = {[a 1, b 1 ], [a 2, b 2 ], [a 1, b 2 ], [a 2, b 1 ]}. Now the set of all maps from X to X denoted by S(X). We have o(s(x)) = 4 4. S (X) s a specal symmetrc nterval semgroup. These structures wll be useful n several applcatons. 57