FINITE NEUTROSOPHIC COMPLEX NUMBERS. W. B. Vasantha Kandasamy Florentin Smarandache

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2 INITE NEUTROSOPHIC COMPLEX NUMBERS W. B. Vsnth Kndsmy lorentn Smrndche ZIP PUBLISHING Oho 11

3 Ths book cn be ordered from: Zp Publshng 1313 Chespeke Ave. Columbus, Oho 31, USA Toll ree: (61) E-ml: Webste: Copyrght 11 by Zp Publshng nd the Authors Peer revewers: Prof. Ion Gon, Deprtment of Algebr, Number Theory nd Logc, Stte Unversty of Kshnev, R. Moldov. Professor Pul P. Wng, Deprtment of Electrcl & Computer Engneerng Prtt School of Engneerng, Duke Unversty, Durhm, NC 778, USA Prof. Mhàly Bencze, Deprtment of Mthemtcs Áprly Ljos College, Brşov, Romn Mny books cn be downloded from the followng Dgtl Lbrry of Scence: ISBN-13: EAN: Prnted n the Unted Sttes of Amerc

4 CONTENTS Prefce 5 Chpter One REAL NEUTROSOPHIC COMPLEX NUMBERS 7 Chpter Two INITE COMPLEX NUMBERS 89 Chpter Three NEUTROSOPHIC COMPLEX MODULO INTEGERS 15 Chpter our APPLICATIONS O COMPLEX NEUTROSOPHIC NUMBERS AND ALGEBRAIC STRUCTURES USING THEM 175 3

5 Chpter ve SUGGESTED PROBLEMS 177 URTHER READING 15 INDEX 17 ABOUT THE AUTHORS

6 PREACE In ths book for the frst tme the uthors ntroduce the noton of rel neutrosophc complex numbers. urther the new noton of fnte complex modulo ntegers s defned. or every C(Z n ) the complex modulo nteger s such tht C(Z n ) re ntroduced nd studed. = n 1. Severl lgebrc structures on urther the noton of complex neutrosophc modulo ntegers s ntroduced. Vector spces nd lner lgebrs re constructed usng these neutrosophc complex modulo ntegers. Ths book s orgnzed nto 5 chpters. The frst chpter ntroduces rel neutrosophc complex numbers. Chpter two ntroduces the noton of fnte complex numbers; lgebrc structures lke groups, rngs etc re defned usng them. Mtrces nd polynomls re constructed usng these fnte complex numbers. 5

7 Chpter three ntroduces the noton of neutrosophc complex modulo ntegers. Algebrc structures usng neutrosophc complex modulo ntegers re bult nd round 9 exmples re gven. Some probble pplctons re suggested n chpter four nd chpter fve suggests round 16 problems some of whch re t reserch level. We thnk Dr. K.Kndsmy for proof redng nd beng extremely supportve. W.B.VASANTHA KANDASAMY LORENTIN SMARANDACHE 6

8 Chpter One REAL NEUTROSOPHIC COMPLEX NUMBERS In ths chpter we for the frst tme we defne the noton of nteger neutrosophc complex numbers, rtonl neutrosophc complex numbers nd rel neutrosophc complex numbers nd derve nterestng propertes relted wth them. Throughout ths chpter Z denotes the set of ntegers, Q the rtonls nd R the rels. I denotes the ndetermncy nd I = I. urther s the complex number nd = 1 or = 1. Also Z I = { + bi, b Z} = N(Z) nd ZI = {I Z}. Smlrly Q I = { + bi, b Q} = N(Q) nd QI = {I Q}. R I = { + bi, b R} = N(R) nd RI = {I R}. or more bout neutrosophy nd the neutrosophc or ndetermnte I refer [9-11, 13-]. Let C( Z I ) = { + bi + c + di, b, c, d Z} denote the nteger complex neutrosophc numbers or nteger neutrosophc complex numbers. If n C Z I = = c = d = then we get pure neutrosophc numbers ZI = {I Z}. If c = d = we get the neutrosophc ntegers { + bi, b Z} = N(Z). If b = d = then we get { + c} the collecton of complex ntegers J. Lkewse P = {di d Z} gve the 7

9 collecton of pure neutrosophc complex ntegers. However di tht s pure neutrosophc collecton ZI s subset of P. Some of the subcollecton wll hve nce lgebrc structure. Thus neutrosophc complex nteger s -tuple { +bi + c + di, b, c, d Z}. We gve opertons on them. Let x = + bi + c + di nd y = m + ni + s + ti be n C( Z I ). Now x + y = ( + bi + c + di) + (m + ni + s + ti) = ( + m) + (b + n)i + (c + s) + (d+t)i. (We cn denote di by di or Id or Id or di) we see x + y s I gn n C( Z I ). We see = + I + + I cts s the ddtve dentty. I I Thus + x = x + = x for every x C( Z I ). In vew of ths we hve the followng theorem. THEOREM 1.1: C( Z I ) = { + bi+ c + di, b, c, d Z} s nteger complex neutrosophc group under ddton. Proof s drect nd hence left s n exercse to the reder. Let x = + bi + c + di nd y = m + ni + t + si be n C( Z I ). To fnd the product xy = ( + bi + c + di) (m + ni + t + si) = m + mbi + mc + mdi + ni + bni + nci + ndi + t + bti + ct + tdi + si + bs + csi + dsi (usng the fct I = I nd = 1). = m + mbi + mc + mdi + ni + bni + nci + ndi + t + bti ct tdi + si + bs csi dsi = (m ct) + (mb + n + bn td cs ds)i + (mc + t) + (md + nc + nd + bt + s + bs)i. Clerly xy C ( Z I ). Now 1 = 1 + I + + I cts s the multplctve I I I dentty. or x 1 = 1 x = x for every x C ( Z I ). Thus C ( Z I ) s monod under multplcton. No element n C ( Z I ) hs nverse wth respect to multplcton. I Hence wthout loss of generlty we cn denote 1 by 1 nd I 1 by. 8

10 THEOREM 1.: C( Z I ) s nteger complex neutrosophc monod or nteger neutrosophc complex monod commuttve monod under multplcton. Proof s smple nd hence left s n exercse to the reder. Also t s esly verfed tht for x, y, z C ( Z I ) x.(y+z) = x.y+x.z nd (x+y)z = x.z + y.z. Thus product dstrbutes over the ddton. In vew of theorems 1. nd 1.1 we hve the followng theorem. THEOREM 1.3: Let S = (C ( Z I ), +, ); = { + bi + c + di where, b, c, d Z}; under ddton + nd multplcton s nteger neutrosophc complex commuttve rng wth unt of nfnte order. Thus S = (C ( Z I ), +, ) s rng. S hs subrngs nd dels. urther C ( Z I ) hs subrngs whch re not dels. Ths s evdent from the followng theorem the proof of whch s left to the reder. THEOREM 1.: Let C ( Z I ) be the nteger complex neutrosophc rng. ) nzi C( Z I ) s nteger neutrosophc subrng of C( Z I ) nd s not n del of C( Z I ) (n = 1,, ) ) nz C ( Z I ) s n nteger subrng of C ( Z I ) whch s not n del of C ( Z I ), (n = 1,, ) ) Let C (Z) = { + b, b Z} C( Z I ), C(Z) s gn complex nteger subrng whch s not n del of C( Z I ). v) Let S = { + bi + c + di, b, c, d nz; < n < } C ( Z I ). S s nteger complex neutrosophc subrng nd S s lso n del of C ( Z I ). The proof of ll these results s smple nd hence s left s n exercse to the reder. We cn defne dels nd lso quotent rngs s n cse of ntegers Z. 9

11 Consder J = { + b + ci + di, b, c, d Z} C ( Z I ) be the del of C ( Z I ). Consder C( Z I ) = {J, 1 + J, + J, I +J, I + J, J, J 1+I+J, 1+I+J, 1++I+J, 1++I+J, 1+I+I+J, +I+I+J, 1++I+I+J, +I+J, +I+J, I+I+J}. Clerly order of P = C( Z I ) s. J We see P s not n ntegrl domn P hs zero dvsors. Lkewse f we consder S = {+b+ci+di, b, c, d, nz} C ( Z I ) S s n del of C ( Z I ). Clerly C( Z I ) s gn not n ntegrl domn nd the S number of elements n C( Z I ) s n. S We cn defne dfferent types of nteger neutrosophc complex rngs usng C( Z I ). DEINITION 1.1: Let C ( Z I ) be the nteger complex neutrosophc rng. Consder S = {(x 1,, x n ) x C ( Z I ); 1 n}; S s gn nteger complex neutrosophc 1 n row mtrx rng. The operton n S s tken component wse nd we get rng. Ths rng s not n ntegrl domns hs zero dvsors. We wll gve some exmples of them. Exmple 1.1: Let S = {(x 1, x, x 3 ) x t = t + b t + c t I + d t I t, b t, c t, d t Z; 1 t 3} be the nteger neutrosophc complex rng. S hs zero dvsors, subrngs nd dels. Exmple 1.: Let V = {(x 1, x,, x 1 ) x t = t + b t + c t I + d t I where t, b t, c t, d t Z; 1 t 1} be the nteger neutrosophc complex 1 1 mtrx rng. V hs zero dvsors, no unts, no dempotents, subrngs nd dels. V s of nfnte order. 1

12 DEINITION 1.: Let S = {( j ) j C ( Z I ); 1, j n} be collecton of n n complex neutrosophc nteger mtrces. S s rng of n n nteger complex neutrosophc rng of nfnte order nd s non commuttve. S hs zero dvsors, unts, dempotents, subrngs nd dels. We gve exmples of them. Exmple 1.3: Let 1 M = C( Z I );1 3 be complex neutrosophc nteger rng. M hs subrngs whch re not dels. or we see 1 N = 1,b1 C( Z I ) M s nteger complex neutrosophc subrng of M whch s only left del of M. Clerly N s not rght del for x b x xb = N. y c d yc yd However c b d x y = x + by cx + dy s n N. Hence N s left del nd not rght del. Consder x y T = x,y C( Z I ) M s subrng but s only rght del s x y c b d = x + yc xb + yd T 11

13 but b x y x y = c d cx dy s not n T hence s only rght del of M. Exmple 1.3: Let M = C( Z I ); be nteger complex neutrosophc rng of mtrces. M s not commuttve. M hs zero dvsors. 1 1 I = 1 1 n M s such tht I s the multplctve dentty n M. Consder b1 b b3 b b5 b6 b7 P = b C( Z I );1 1 M b8 b 9 b 1 s n nteger complex neutrosophc subrng whch s not left del or rght del of M. Consder 1 3 T = C( Z I );1 1 M

14 s n nteger complex neutrosophc subrng only nd not left del or rght del of M. Now we cn proceed onto defne polynoml nteger complex neutrosophc rng. DEINITION 1.: Let V = x C( Z I ) = be the collecton of ll polynomls n the vrble x wth coeffcents from the nteger complex neutrosophc ntegrl rng C ( Z I ) wth the followng type of ddton nd multplcton. If p (x) = + 1 x + + n x n nd q(x) = b + b 1 x + + b n x n where, b C ( Z I ); 1 n; re n V then p (x) + q (x)= ( + 1 x + + n x n ) + (b + b 1 x + + b n x n ) = ( + b ) + ( 1 + b 1 )x + + ( n + b n )x n V. The = + x + + x n s the zero nteger complex neutrosophc polynoml n V. Now p(x). q(x) = b + ( b b ) x + + n b n x n s n C ( Z I ). 1 = 1+ x + + x n n V s such tht p (x). 1 = 1. p (x) = p (x). (V, +,.) s defned s the nteger complex neutrosophc polynoml rng. We just enumerte some of the propertes enjoyed by V. () V s commuttve rng wth unt. () V s n nfnte rng. We cn defne rreducble polynomls n V s n cse of usul polynomls. p (x) = x V we see p (x) s rreducble n V. q (x) = x 3 3 V s lso rreducble n V. q (x) = x 1 5 s lso rreducble n V. p (x) = x + s rreducble n V. 13

15 Thus n V we cn defne reducblty nd rreducblty n polynomls n V. Let p(x) V f p(x) = (x 1 ) (x t ) where C( Z I ); 1 t then p(x) s reducble lnerly n unque mnner except for the order n whch s occur. It s nfct not n esy tsk to defne reltvely prme or gretest common dvsor of two polynomls wth coeffcents from C ( Z I ). or t s stll dffcult to defne g c d of two elements n C ( Z I ). or f = 5 + 3I nd b = (7 + 5I) we cn sy g c d (, b) = 1, f = 3 + I nd b = I then lso g c d (, b) =1. If = I + 9I nd b = 1I then g c d (, b) = 3 nd so on. So t s by lookng or workng wth, b n C ( Z I ) we cn fnd g c d. Now hvng seen the problem we cn not put ny order on C ( Z I ). or consder nd I we cnnot order them for s the complex number nd I s n ndetermnte so no relton cn be obtned between them. Lkewse 1 + nd I nd so on. Concept of reducblty nd rreducblty s n esy tsk but other concepts to be obtned n cse of neutrosophc complex nteger polynomls s dffcult tsk. Thus C( Z I )[x] = x C( Q I ) = s commuttve ntegrl domn. Let p(x) nd q(x) C( Z I )[x], we cn defne degree of p (x) s the hghest power of x n p(x) wth non zero coeffcents from C( Z I ). So f deg(p(x)) = n nd deg q(x) = m nd f n < m then we cn dvde q(x) by p(x) nd fnd where deg s(x) < n. q(x) p(x) = r (x) + s(x) p(x) 1

16 Ths dvson lso s crred out s n cse of usul polynomls but due to the presence of four tuples the dvson process s not smple. Z[x] C ( Z I ) [x] = S. Thus Z [x] s only subrng nd Z[x] s n ntegrl domn n C ( Z I ) [x] = S. Lkewse ZI[x] C ( Z I ) [x] s gn n ntegrl domn whch s subrng. Consder P = { + bi, b Z} C( Z I ) [x], P s gn subrng whch s not n del of C( Z I ) [x]. Also C (Z) = { + b, b Z} s gn subrng of C( Z I )[x]. urther C(Z)[x] = x C(Z) C ( Z I ) [x] = s only subrng of C ( Z I ) [x] whch s not n del. Lkewse C(ZI)[x] = x C(ZI) ; = + b, b ZI} S = s only subrng of S nd s lso n del of S. Severl such propertes enjoyed by C ( Z I ) [x] cn be derved wthout ny dffculty. We cn lso defne the noton of prme del s n cse of C ( Z I ) [x]. Now we cn lso defne semgroups usng C ( Z I ). Consder 1 C ( Z I ); 1 n} = H; n 1 n H s group under ddton but multplcton cnnot be defned on H. So H s not rng but only semgroup. Thus ny collecton of m n mtrces wth entres from C ( Z I ) (m n) s only n beln group under ddton nd s not rng s multplcton cnnot be defned on tht collecton. 15

17 We cn replce Z by Q then we get C( Z I ) to be rtonl complex neutrosophc numbers. Also C( Z I ) C( Q I ). C( Q I ) = { + b + ci + di, b, c, d Q} s rng. Infct C ( Q I ) hs no zero dvsors. or f we tke x = 6 + 3I + I nd y = + b + ci + di n C ( Q I ). xy = (6 + 3I + I) ( + b + ci + di) = 6 + 3I + I 6b b 3bI bi 6cI + ci 3cI + ci 6dI di 3dI di = (6 b) + I ( + 6b) + ( 3 b + 6c 3c d d)i + ( 3b + c + c + 6d 3d) = 6 = b + 6b = b = 3 = 3b ths s possble only when = b = 3c 6d = c = d 6c + 3d = d = c So c = d =. Thus = b = c = d =. But n generl C( Q I ) s not feld. Ths feld contns subfelds lke Q, S = { + b, b Q} C ( Q I ) contns lso subrngs. We cn buld lgebrc structures usng C ( Q I ). We cll C ( Q I ) s the rtonl complex neutrosophc lke feld. C ( Q I ) s of chrcterstc zero C ( Q I ) s not prme lke feld for t hs subfelds of chrcterstc zero. C ( Q I )[x] s defned s the neutrosophc complex rtonl polynoml; C( Q I ) [x] = x C( Q I ). = C ( Q I ) s not feld, t s only n ntegrl domn we cn derve the polynoml propertes relted wth rtonl complex neutrosophc polynomls n C ( Q I ) [x]. 16

18 C( Q I ) [x] cn hve dels. Now consder T = {(x 1, x,, x n ) x C ( Q I ); 1 n}; T s rtonl complex neutrosophc 1 n mtrx. T s only rng for T contns zero dvsors but T hs no dempotents;t hs dels for tke P = {(x 1, x, x 3,,, ) x C ( Q I ); 1 3} T s subrng s well s n del of T. T hs severl dels, T lso hs subrngs whch re not dels. or tke S = {(x 1, x, x 3,, x n ) x C ( Z I ); 1 n} T; S s subrng of T nd s not n del of T. We cn hve severl subrngs of T whch re not dels of T. Now we cn defne M = {A = ( j ) A s n n rtonl complex neutrosophc mtrx wth j C ( Q I ); 1, j n} to be the n n rtonl complex neutrosophc mtrx rng. M lso hs zero dvsors, unts, dels, nd subrngs. or consder N = {collecton of ll upper trngulr n n mtrces wth elements from C ( Q I )} M; N s subrng of M nd N s not n del of M. We hve T = {ll dgonl n n mtrces wth entres from C ( Q I )} M; T s n del of M. All usul propertes cn be derved wth pproprte modfctons. Now f we replce Q by R we get C ( R I ) to be the rel complex neutrosophc rng. C ( R I ) s not feld clled the lke feld of rel complex neutrosophc numbers. C ( R I ) s not prme feld. It hs subfelds nd subrngs whch re not subfelds. All propertes enjoyed by C( Q I ) cn lso be derved for C( R I ). We see C ( R I ) C ( Q I ) C ( Z I ). We construct polynoml rng wth rel complex neutrosophc coeffcents nd 1 n mtrx rng wth rel complex neutrosophc mtrces. Lkewse the n n rel complex neutrosophc mtrx rng cn lso be constructed. The ltter two wll hve zero dvsors nd unts where s the frst rng hs no zero dvsors t s n ntegrl domn. 17

19 Now we hve seen rel complex neutrosophc lke feld, C ( R I ). We proceed onto defne vector spces, set vector spces, group vector spces of complex neutrosophc numbers. DEINITION 1.: Let V be ddtve beln group of complex neutrosophc numbers. Q be the feld. If V s vector spce over Q then defne V to be ordnry complex neutrosophc vector spce over the feld Q. We wll gve exmples of them. Exmple 1.5: Let 1 3 V = 5 6 C( Q I ); be n ordnry complex neutrosophc vector spce over Q. Exmple 1.6: Let 1 M = C( Q I );1 3 be the ordnry complex neutrosophc vector spce over Q. Exmple 1.7: Let 1 3 P = C( Q I ); be the ordnry complex neutrosophc vector spce over Q. We cn s n cse of usul vector spces defne subspces nd bss of P over Q. 18

20 However t s pertnent to menton here tht we cn hve other types of vector spces defned dependng on the feld we choose. DEINITION 1.5: Let V be the complex neutrosophc ddtve beln group. Tke = { + b, b Q; = 1}; f V s vector spce over the complex feld ; then we cll V to be complex - complex neutrosophc vector spce. We wll gve exmples of them. Exmple 1.8: Let 1 3 V = C( Q I );1 8 be complex - complex neutrosophc vector spce over the feld = { + b, b Q}. Tke 1 3 W = C( Q I );1 s complex - complex neutrosophc vector subspce of V over. Infct V hs severl such subspces. Tke 1 W 1 = C( Q I );1 3 V, 3 complex - complex neutrosophc vector subspce of V over the rtonl complex feld. Suppose W = 1, C( Q I ) V 1 s complex - complex neutrosophc vector subspce of V over. 19

21 Consder W 3 = 1 3 1,,3 C( Q I ) V; W 3 s complex - complex neutrosophc vector subspce of V over. Clerly V = W 1 W W 3 = W 1 + W + W 3. Thus V s drect sum of subspces. Exmple 1.9: Let V = C( Q I ); be complex - complex neutrosophc vector spce over the complex feld = { + b, b Q}. V hs subspces nd V cn be wrtten s drect sum / unon of subspces of V over. Now we cn defne complex neutrosophc - neutrosophc lke vector spce or neutrosophc - neutrosophc vector spce over the neutrosophc lke feld Q I or R I. DEINITION 1.6: Let V be n ddtve beln group of complex neutrosophc numbers. Let = Q I be the neutrosophc lke feld of rtonls. If V s lke vector spce over then we defne V to be neutrosophc - neutrosophc complex lke vector spce over the feld (complex neutrosophc - neutrosophc vector spce over the feld or neutrosophc complex neutrosophc vector spce over the feld ). We wll gve exmples of ths stuton.

22 Exmple 1.1: Let 1 3 V = 5 6 C( Q I ); be neutrosophc complex neutrosophc lke vector spce over the neutrosophc lke feld = Q I. Exmple 1.11: Let M = C( Q I ); be neutrosophc - neutrosophc complex lke vector spce over the neutrosophc lke feld Q I =. Tke P 1 = 1 3 C( Q I );1 V, 1 3 P = C( Q I );1 V, 1

23 P 3 = C( Q I );1 V 1 5 nd P = C( Q I );1 V 1 3 be subspce of V. Clerly V = P 1 + P + P 3 + P nd P P j = () f j, so V s drect sum of subspces of V over = Q I. Consder V 1 = V = 1 C( Q I ); C( Q I );1 6 6 V, V, V 3 = 1 3 C( Q I );1 6 V, 5

24 1 V = C( Q I );1 7 V, nd 1 3 V 5 = C( Q I );1 7 V s such tht V = V 1 V V 3 V V 5 but V V j () f j so V s only pseudo drect sum of the subspces of V over. Now we hve defned neutrosophc complex - neutrosophc lke vector spces over the neutrosophc lke feld. We now proceed onto defne specl complex neutrosophc lke vector spce over the complex neutrosophc lke feld. DEINITION 1.7: Let V be n beln group under ddton of complex neutrosophc numbers. Let = C ( Q I ) be the complex - neutrosophc lke feld of rtonls; f V s vector spce over then we defne V to be specl complex neutrosophc lke vector spce over the complex neutrosophc rtonl lke feld = C ( Q I ). We wll gve exmples of them. Exmple 1.1: Let 1 V = 15 where C( Q I ); 1 15} be the specl complex neutrosophc lke vector spce over the complex neutrosophc lke feld of rtonls C( Q I ) =. 3

25 It s esly verfed V hs subspces. The dmenson of V over s 15. or tke B =,,,...,, V 1 1 s B s bss of V over. Exmple 1.13: Let V = C( Q I ); be specl complex neutrosophc lke vector spce over the feld = C ( Q I ). Now we cn defne lke subfeld subspce of vector spce. DEINITION 1.8: Let V be n ddtve beln group of complex neutrosophc numbers. V be specl complex neutrosophc vector spce over the complex neutrosophc lke feld = C ( Q I ) = { + b + ci + di, b, c, d Q}. Let W V, W lso proper subgroup of V nd K. K the neutrosophc lke feld Q I C ( Q I ) =. If W s vector spce over K then we defne W to be neutrosophc subfeld complex neutrosophc vector subspce of V over the neutrosophc lke subfeld K of. We wll gve exmples of ths stuton.

26 Exmple 1.1: Let V = C( Q I ); be specl complex neutrosophc vector spce over the complex neutrosophc lke feld = C ( Q I ). Nothng s lost f we sy neutrosophc feld lso for we hve defned so but n ndetermnte or neutrosophc feld need not hve rel feld structure lke neutrosophc group s not group yet we cll t group. Tke 1 3 W = C( Q I );1 6 V; 5 6 tke K = Q I = C ( Q I ); W s neutrosophc specl complex neutrosophc vector subspce of V over the neutrosophc lke subfeld Q I of = C ( Q I ). Consder 1 M = 3 C( Q I );1 5 V 5 tke K = { + b, b Q} = C ( Q I ) complex subfeld of C ( Q I ). W s specl complex neutrosophc complex subvector spce of V over the rtonl complex subfeld K of. 5

27 Suppose 1 3 S = 5 C( Q I ); V. Tke Q C ( Q I ) = the feld of rtonls s subfeld of. S s ordnry specl neutrosophc complex vector subspce of V over the rtonl subfeld Q of = C ( Q I ). All propertes cn be derved for vector spce over C ( Q I ). It s nterestng to see tht only these specl neutrosophc complex vector spces hs mny propertes whch n generl s not true over the usul feld Q. THEOREM 1.5: Let V be specl neutrosophc complex vector spce over C ( Q I ). V hs only subfelds over whch vector subspces cn be defned. When we sy ths t s evdent from the exmple 1.1, hence left s n exercse to the reder. THEOREM 1.6: Let V be n ordnry neutrosophc complex vector spce over the feld Q. V hs no subfeld vector subspce. Proof esly follows from the fct Q s prme feld. THEOREM 1.7: Let V be neutrosophc complex neutrosophc vector spce over the neutrosophc feld Q I =. V hs only one subfeld over whch vector subspces cn be defned. THEOREM 1.8: Let V be complex neutrosophc complex vector spce over the rtonl complex feld = { + b, b Q}. V hs only one subfeld over whch subvector spces cn be defned. 6

28 Proof follows from the fct Q. We see Q Q I C ( Q I ), Q C (Q) = { + b, b Q} C ( Q I ). These spces behve n very dfferent wy whch s evdent from the followng exmple. Exmple 1.15: Let 1 3 V = 5 6 C( Q I ); be specl complex neutrosophc vector spce over the complex neutrosophc lke feld = C ( Q I ). It s esly verfed V s of dmenson 9 over = C ( Q I ). However V hs specl complex neutrosophc subspces of dmensons 1,, 3,, 5, 6, 7 nd 8. Now consder 1 W = C( Q I );1 V. W s specl subfeld neutrosophc complex neutrosophc vector subspce of V over the neutrosophc subfeld Q I of. Clerly dmenson of W over Q I s not fnte. Suppose W s consdered s specl neutrosophc complex vector subspce of V over = C ( Q I ) then dmenson of W over s four. W s specl complex neutrosophc complex vector subspce over the rtonl complex feld K = { + b, b Q} C ( Q I ) whch s lso of nfnte dmenson over K. W s specl neutrosophc complex ordnry vector subspce over the rtonl feld Q s lso of nfnte dmenson over the rtonl subfeld Q of. 7

29 Consder T = 1 3 Q;1 5 5 V. T s specl neutrosophc complex ordnry vector subspce of V over Q dmenson 5. Clerly T s not defned over the subfeld, Q I of C(Q) or C( Q I ). Consder 1 S = 3 Q I ;1 V be specl neutrosophc-neutrosophc complex vector subspce of V over the neutrosophc rtonl subfeld Q I. S s of dmenson four over Q I. S s lso specl neutrosophc complex ordnry vector subspce of V over the rtonl feld Q nd S s of nfnte dmenson over Q. Clerly S s not vector subspce over C(Q) = { + b, b Q} or C( Q I ). Consder 1 3 P = 5 C(Q) 6 = { + b, b Q}, 1 6} V, P s specl complex neutrosophc complex vector subspce of V over the rtonl complex feld C(Q) of dmenson 6. P s lso specl complex neutrosophc ordnry vector subspce over the feld of rtonls Q of nfnte dmenson. Clerly P s not specl complex neutrosophc complex vector subspce of V over Q I or C ( Q I ). As t s not defned over the two felds. 8

30 Thus we hve seen severl of the propertes bout specl neutrosophc complex vector spces defned over C ( Q I ). We now proceed onto defne the lner lgebr structures. DEINITION 1.9: Let V be ordnry complex neutrosophc vector spce over the rtonls Q. If on V product. cn be defned nd V s comptble wth respect to the product., then we cll V to be ordnry complex neutrosophc lner lgebr over the rtonl feld Q. We provde exmples of them. Exmple 1.16: Let V = 1 3 C( Q I );1 be n ordnry complex neutrosophc lner lgebr over Q. V s of nfnte dmenson over Q. V hs lner sublgebrs. It s nterestng to notce V s lso vector spce but ll ordnry complex neutrosophc vector spces over Q need not n generl be lner lgebrs. In vew of ths we hve the followng theorem. THEOREM 1.9: Let V be n ordnry neutrosophc complex lner lgebr over the rtonls Q, then V s n ordnry neutrosophc complex vector spce. If V s n ordnry neutrosophc complex vector spce over Q then V n generl s not n ordnry neutrosophc complex lner lgebr over Q. The proof s strght forwrd hence left s n exercse for the reder. On smlr lnes we cn defne neutrosophc - complex neutrosophc lner lgebr over Q I, neutrosophc complex - complex lner lgebr over C(Q) = { + b, b Q} nd specl neutrosophc complex lner lgebr over C( Q I ). We gve only exmples of them s the defnton s mtter of routne. 9

31 Exmple 1.17: Let V = x C( Q I ) = be neutrosophc complex neutrosophc lner lgebr over the neutrosophc feld = Q I. Exmple 1.18: Let M = {All 5 5 upper trngulr mtrces wth entres from C ( Q I )} be neutrosophc complex neutrosophc lner lgebr over the neutrosophc feld Q I = Exmple 1.19: Let P = {ll 1 1 mtrces wth complex neutrosophc entres from C( Q I )} be neutrosophc complex neutrosophc lner lgebr over the neutrosophc feld = Q I. M = {ll 1 1 upper trngulr mtrces wth complex neutrosophc entres from C( Q I )} P s the neutrosophc complex neutrosophc lner sublgebr of P over the feld = Q I. Ths spce s of nfnte dmenson over Q I =. Exmple 1.: Let P = x C( Q I ) = be complex neutrosophc complex lner lgebr over the complex rtonl feld C(Q) = { + b, b Q}. Exmple 1.1: Let V = C( Z I ); be complex neutrosophc complex lner lgebr over the rtonl complex feld C(Q) = { + b, b Q}. 3

32 Exmple 1.: Let M = {ll 8 8 lower trngulr complex neutrosophc mtrces wth entres from C ( Q I )} be complex neutrosophc complex lner lgebr over the complex feld C(Q) = { + b, b Q}. Exmple 1.3: Let 1 3 T = 5 6 C( Q I ); be n ordnry neutrosophc complex lner lgebr over the rtonl feld Q. Exmple 1.: Let A = {ll 1 1 upper trngulr mtrces wth complex neutrosophc entres from the complex neutrosophc feld} be n ordnry neutrosophc complex lner lgebr over the feld of rtonls Q. Exmple 1.5: Let B = x C( Q I ) = be specl neutrosophc complex lner lgebr over the neutrosophc complex feld C ( Q I ). Exmple 1.6: Let 1 3 C = 5 C( Q I );1 6 6 be specl neutrosophc complex lner lgebr over the neutrosophc complex feld C ( Q I ). Clerly C s of dmenson 6 over C ( Q I ). We cn derve lmost ll propertes of vector spces n cse of lner lgebrs wth smple pproprte modfctons. 31

33 We cn lso defne the noton of lner trnsformton nd lner opertors. We cn defne lner trnsformton T of neutrosophc complex vector spces from V to W only f V nd W re defned over the sme feld. Also T (I) = I s bsc crter for the trnsformton for the ndetermnncy cnnot be mpped on to ny other element. Lkewse f W s replced by V ths lner trnsformton T of V to V becomes the lner opertor. All propertes ssocted wth lner trnsformton nd lner opertors of neutrosophc complex vector spces cn be esly derved n cse of these opertors. We cn now defne the chrcterstc vlues nd chrcterstc vectors s n cse of complex neutrosophc vector spces / lner lgebrs defned over the felds. DEINITION 1.1: Let V be specl complex neutrosophc vector spce over the complex neutrosophc feld C ( Q I ). Let T be specl lner opertor on V. A complex neutrosophc vlue of T s sclr c n C ( Q I ) so tht there s non zero neutrosophc complex vector α n V wth Tα = cα. If c s the specl chrcterstc vlue of T then ) ny α such tht Tα = cα s clled the chrcterstc neutrosophc complex vector of T ssocted wth c. ) The collecton of ll α such tht Tα = cα s clled the neutrosophc complex chrcterstc spce ssocted wth c. If the complex neutrosophc feld C( Q I ) s replced by Q I or C(Q) = { + b, b Q} or Q we get the chrcterstc vlue c s neutrosophc number + bi or c + d or respectvely nd the ssocted chrcterstc spce of them would be neutrosophc complex neutrosophc subspce or complex neutrosophc complex subspce or ordnry complex neutrosophc subspce respectvely. The followng theorem s left n exercse for the reder to prove. 3

34 THEOREM 1.1: Let T be complex neutrosophc lner opertor on fnte dmensonl specl (ordnry or complex or neutrosophc) complex neutrosophc vector spce V nd let c be sclr n C( Q I ) (or Q or C(Q) or Q I respectvely), The followng re equvlent ) c s the chrcterstc vlue of T. ) The opertor (T ci) s sngulr. ) det (T ci) =. We cn defne dgonlzble lner opertor s n cse of other lner opertor. Also we wll show by n exmple how to fnd the chrcterstc polynoml n cse of specl (ordnry or complex or neutrosophc) mtrces. Let M = 1 + b1 + c1i + d1i + b + ci + di 3 + b3 + c3i + d3i + b + ci + di, be mtrx wth entres over the feld C ( Q I ) such tht the mtrx (M ci ) s non nvertble. All results relted wth lner opertors of specl (ordnry or complex or neutrosophc) vector spces cn be derved s n cse of usul vector spces wth smple pproprte modfctons. Now we proceed onto defne the concept of lner functonls. We cn defne four types of lner functonls. Let V be specl neutrosophc complex vector spce over the neutrosophc complex feld = C( Q I ). The specl lner functonl on V s mp ( lner trnsformton) f : V such tht (c α + β ) = cf (α) + f (β) α, β V nd C C ( Q I ). We wll frst llustrte ths by n exmple. 33

35 Exmple 1.7: Let V = C( Q I );1 6 be specl neutrosophc complex vector spce over the neutrosophc complex feld = C( Q I ). Defne f : V by f = 6. = Clerly f s specl lner functonl on V. If V s neutrosophc complex neutrosophc vector spce over the neutrosophc feld K = Q I. We defne f : V K nd f (v) Q I tht s only neutrosophc number. Exmple 1.8: Let 1 3 V = 5 6 C( Q I );1 6 be neutrosophc complex neutrosophc vector spce over the neutrosophc feld Q I = K. Defne f : V K by f 1 = ( ) + (c c 16 )I 6 3

36 where 1 = 11 + b 11 + c 11 I + d 11 I = 1 + b 1 + c 1 I + d 1 I nd 6 = 16 + b 16 + c 16 I + d 16 I. Clerly 1 f = ( ) + (c c 16 )I 6 s n Q I s neutrosophc lner functonl on V. Exmple 1.9: Let V = {( 1,, 3 ) C ( Q I ); 1 3} be complex neutrosophc complex vector spce over the rtonl complex feld K = C(Q) = { + b, b Q}. Let f : V K defned by f ( 1,, 3 ) = ( ) + (b 11 + b 1 + b 13 ) where 1 = 11 + b 11 + c 11 I + d 11 I = 1 + b 1 + c 1 I + d 1 I nd 3 = 13 + b 13 + c 13 I + d 13 I where j, b j, c j nd d j re n Q; 1 j 3. Clerly f s complex lner functonl on V. Now we proceed onto gve n exmple of ordnry lner functonl on V. Exmple 1.3: Let V = 1 3 j = 1 + b 1 + c 1 I + d 1 I where 1, b 1, c 1, d 1 Q; 1 3 nd 1 j } be n ordnry neutrosophc complex vector spce over the rtonl feld Q. Defne f : V Q by 35

37 1 f = ; f s n ordnry lner functonl on V. Now hvng seen the defntons of lner functonls nterested reder cn derve ll propertes relted wth lner functonls wth pproprte chnges. We cn now defne set neutrosophc complex vector spces, semgroup neutrosophc complex vector spces nd group neutrosophc complex vector spces. Also the correspondng lner lgebrs. DEINITION 1.11: Let V C ( Q I ) be proper subset of complex neutrosophc rtonls. S Q be subset of S. We defne V to set complex neutrosophc vector spce over the set S Q f for ll v V nd s S, vs nd sv V. We gve exmples of ths stuton. Exmple 1.31: Let V = {( 1,, 3 ), , x C( Q I ); = be set vector spce of neutrosophc complex rtonls over Z Q or set complex neutrosophc rtonl vector spce over the set Z. Exmple 1.3: Let V =, 3 5 C ( Q I ); 1 5} be set complex neutrosophc vector spce over the set S = {, 1,, 3,, 1, 17, -5, -9, -3} Q. 36

38 Exmple 1.33: Let 1 8 V = x,( 1,, 3,..., ), C ( Q I ); = 1 } be set neutrosophc complex vector spce over the set S = {, 1, 1}. Now hvng seen exmples of set neutrosophc complex vector spces we now proceed onto defne set neutrosophc complex vector subspces of V over the set S. Let V be set complex neutrosophc vector spce over the set S. Suppose W V nd f W tself s set complex neutrosophc vector spce over the set S then we defne W to be set complex neutrosophc vector subspce of V over S. We wll llustrte ths stuton by some exmples. Exmple 1.3: Let 1 3 V = ( 1,, 3 ), 3, x C( Q I ); 5} = 5 be set vector spce of neutrosophc complex rtonls over the set S = {, 1}. Consder 1 W = (, 1,), 1,,3 C( Q I ) V, 3 37

39 W s set vector subspce of neutrosophc complex rtonls over the set S ={, 1}. Tke 1 3 M = ( 1,, 3), 3 x C ( Z I ) = 5 C( Q I ); 5} V s set complex neutrosophc vector subspce of V over the set S = {, 1}. Exmple 1.35: Let V = x,, = C ( Q I ); } be set neutrosophc complex vector spce over the set { 5,,, 1, 3, 8, 1, } S. Consder W = x,, = j, C( Q I ); 1, j = 1,, 3,, 5, 6, 7, 8, 9, 1, 1,,, 3} V s set neutrosophc complex vector subspce of V over the set S. Now s n cse of usul set vector spces we cn derve ll the relted propertes we cn lso defne the noton of subset 38

40 neutrosophc complex vector subspce over subset, whch s smple nd left s n exercse to the reder. We gve exmples of ths structure. Exmple 1.36: Let V = , 5 6, C ( Q I ); 1 16} be set neutrosophc complex vector spce over the set S = 3Z 5Z 7Z 13Z. Consder P = , 3, C ( Q I ); 1 8} V; P s subset neutrosophc complex vector subspce of V over the subset T = 3Z 13Z S. Also M = 1 1 3,, C ( Q I ); 1 } V, M s subset neutrosophc complex vector subspce of V over the set T = 3Z 5Z S. 39

41 Exmple 1.37: Let V = = 3 x,,,(,,, ) 11 1 C ( Q I ); 9}be set neutrosophc complex vector spce over the set S = 3Z + 5Z 7Z +. Consder 1 1 M = x,,,( 1,,,) = 3 C ( Q I ); } V; s subset neutrosophc complex vector subspce of V over the subset T = {3Z + 5Z} S. Tke P = x,,,(,, 1, ) = C ( Q I ); 1} V, s subset vector complex neutrosophc subspce of V over the subset T = 7Z + S. We cn defne set lner trnsformton, set lner opertor nd set bss of set neutrosophc complex vector spce over the set S, whch s left s n exercse s t cn be crred out s mtter of routne. We cn lso defne specl set neutrosophc complex vector spce over the complex neutrosophc subset of C( Q I ).

42 Exmple 1.38: Let V = ,, 5 6 C( Q I ); be set neutrosophc of complex neutrosophc vector over set S = {1,, 1 + I, 3 1, I I} C( Q I ). We observe S Q so we cll such vector spces s specl set vector spces. DEINITION 1.1: Let V be set vector spce of complex neutrosophc rtonls (set complex neutrosophc rtonls vector spce) over set S C ( Q I ) nd S Q then we defne V to be specl set vector spce of complex neutrosophc rtonls over the set S f vs nd sv V for ll s S nd v V. We wll llustrte ths by some exmples. Exmple 1.39: Let V = ,,( 1,,..., 1 ) C( Q I );1 1} be specl set complex neutrosophc rtonl vector spce over the set S = C ( Z I ) C ( Q I ). Exmple 1.: Let M = 5 6,,

43 C( Q I ); 1 18} be specl set neutrosophc complex vector spce over the set S = C( 3Z I ) C( Q I ). Exmple 1.1: Let V = ( 1,,..., 9 ), , x = C( Q I ); 1} be specl set complex neutrosophc vector spce over the set C( 3Z I ). Exmple 1.: Let M =, 5 6, x, = C( Q I ); 5} be specl set complex neutrosophc vector spce over the set S = {C( 5Z I ) C(13Z I )}.Tke P = 1 1 x,, C( < Q I > ); 1 = 3 V, P s specl set neutrosophc complex vector subspce of V over S. Consder W = 1 C ( Q I ); } 1, x, 3 = 3 5

44 V, W s specl set complex neutrosophc vector subspce of V over the set S. Consder B =,, x C ( Q I ); = 3 3 5} V, tke T = {C ( 5Z I ) C( 39Z I )} S; we see B s specl subset complex neutrosophc vector subspce of V over the subset T of S. Now hvng seen exmples of specl set subspces nd specl subset vector subspces of complex neutrosophc rtonls we proceed onto defne the noton of semgroup complex neutrosophc vector spce nd specl semgroup complex neutrosophc vector spce. Just we menton n cse of specl set neutrosophc complex vector spce lso one cn defne specl set lner trnsformtons provded both the specl set vector spces re defned over the sme set of complex neutrosophc numbers. urther the bss, drect sum of subspce nd other propertes cn be esly derved s mtter of routne. All these work s left s exercses to the reder. DEINITION 1.13: Let V be ny subset of complex neutrosophc numbers nd S be ny ddtve semgroup wth zero. We cll V to be semgroup neutrosophc complex vector spce over S f the followng condtons hold good. ) vs = sv V for ll s S nd v V. ).v = V for ll v V nd S; V s zero vector. ) (s 1 + s ) v = s 1 v + s v for ll s 1, s S nd v V. 3

45 We wll frst llustrte ths stuton by some exmples. Exmple 1.3: Let V = ,, C( Q I ); 1 3} be semgroup neutrosophc complex vector spce over the semgroup S = Z + {}. Exmple 1.: Let = M = x,,, C( Q I ); } be semgroup neutrosophc complex vector spce over the ddtve semgroup S = 5Z + {}. Exmple 1.5: Let P = ,, C ( Q I ); 1 1} be semgroup neutrosophc complex vector spce over the semgroup S = Z + {}.

46 Consder V = 1 1 1,, ,, 3, 9, 1 C ( Q I } P; V s semgroup neutrosophc complex vector subspce of P over the semgroup S. Also B =, , C ( Q I } B s subsemgroup neutrosophc complex vector subspce of P over the subsemgroup A = 8Z + {} of the semgroup S. Here lso ll propertes of semgroup lner trnsformton of vector spces cn be obtned provded they re defned over the sme, semgroup, semgroup lner opertor nd bss cn be defned s n cse of usul semgroup vector spces. Exmple 1.6: Let V = , , C( Q I ); 1 15} be semgroup complex neutrosophc vector spce over the semgroup S = 3Z + {}. 5

47 Consder W 1 = 1 3 C ( Q I ); 1 } V, W = nd W 3 = 9 1 C( Q I );1 15} V C ( Q I ); 1 1} V be semgroup complex neutrosophc vector subspces of V over the semgroup S = 3Z + {}. It s esly seen V = W 1 + W + W 3 nd W W j = φ; 1, j 3. Thus V s drect sum of semgroup complex neutrosophc vector subspces of V over S. Let W 1 = 1 3 1, 3 C ( Q I ); 1 } V, W = , , C ( Q I ); 1 15} V nd 6

48 W 3 = , C ( Q I ); 1 1} V; clerly W 1 + W + W 3 = V nd W W j φ f j; 1, j 3. Thus V s only pseudo drect unon of subspces. We defne specl semgroup complex vector s follows. DEINITION 1.1: Let V be semgroup neutrosophc complex vector spce over the complex neutrosophc ddtve semgroup S. Then we defne V to be specl semgroup neutrosophc complex vector spce over the semgroup S. We wll llustrte ths stuton by some exmples. Exmple 1.7: Let V =,, C( Q I ); 1 } be specl semgroup neutrosophc complex vector spce over the complex neutrosophc ddtve semgroup S = C ( 3Z I ). Exmple 1.8: Let M =,, x =

49 C( Q I ); 5} be specl semgroup complex neutrosophc vector spce over the semgroup under ddton S = C( 5Z I ). Exmple 1.9: Let M = x, 5 6, C( Q I ); 3 = be specl semgroup complex neutrosophc vector spce over the neutrosophc complex semgroup S = C( 1Z I ) under ddton. Tke V = x, 5 C( Q I ); 6 M = 6 be specl semgroup complex neutrosophc vector subspce of M over the neutrosophc complex semgroup S. Tke 1 6 P = x, C( Q I ); 16 M = 16 be specl subsemgroup complex neutrosophc subvector spce of M over the subsemgroup T = C ( Z I ) S under ddton. We cn lso wrte M s drect sum of subspces s well s pseudo drect sum of specl semgroup vector subspces. Now we proceed onto defne the noton of set lner lgebr, specl set lner lgebr, semgroup lner lgebr nd specl semgroup lner lgebr usng complex neutrosophc rtonls. 8

50 DEINITION 1.15: Let V be set neutrosophc complex vector spce over the set S. We defne V to be set complex neutrosophc lner lgebr over the set S f s ( + b) = s + sb for ll, b V nd s S. We gve exmples of them. Exmple 1.5: Let 1 3 V = C( Q I ); be set complex neutrosophc lner lgebr over the set S = (5Z 7Z). Exmple 1.51: Let 1 M = C( Q I );1, 3 be set complex neutrosophc lner lgebr over the set S = 3Z 7Z 5Z. Exmple 1.5: Let M = C( Q I ); be set neutrosophc complex lner lgebr over the set S = 5Z Z 17Z W= 1, 3,..., C( Q I ) 1... M s set neutrosophc complex lner sublgebr of M over the set S. Tke R = 1,b1 C( Q I ) b1 b 1... b M; 1 9

51 R s set neutrosophc complex lner lgebr over the set S = 5Z Z 17Z. Consder C( Q I ); A = M be subset complex neutrosophc lner sublgebr of M over the subset T = 5Z Z S. Suppose B = C( Q I ); M; B s subset neutrosophc complex lner sublgebr of M over the subset R = 17Z S. Exmple 1.53: Let M = C( Q I );1 18 be set neutrosophc complex lner lgebr over the set S = 3Z + {}. Consder W 1 = C( Q I );1 6 5

52 proper subset of M, 1 3 W = 5 6 C( Q I );1 6 M, W 3 = 1,,3 C( Q I ) M 1 3 nd W = C( Q I );1 3 M 1 3 be set neutrosophc complex lner sublgebrs of M over the set S. Clerly M = W 1 + W + W 3 + W nd W W j = () f j; 1, j. Thus M s the drect sum of sublner lgebrs of M. Now we cn lso defne specl set lner lgebr of complex neutrosophc numbers. We wll gve only exmples of them nd ther substructures s t s mtter of routne to defne them. 51

53 Exmple 1.5: Let V = C( Q I ); be specl set lner lgebr of complex neutrosophc rtonls over the set S = C ( Z I ). Exmple 1.55: Let M = C( Q I ); be specl set neutrosophc complex lner lgebr over the set S = C ( 3Z I ) C ( 5Z I ). Exmple 1.56: Let V = C( Q I );1 8 be specl set complex neutrosophc lner lgebr of V over the set S = C ( 3Z I ) C ( 5Z I ). Tke M = C( Q I );1 6 V be specl set complex neutrosophc lner sublgebr of V over the set S. 5

54 Consder 1... T = C( Q I );1... V, be specl set complex neutrosophc lner sublgebr of V over S. Tke... A = C( Q I ); V nd subset T = C ( 3Z I ) S. A s subset specl neutrosophc complex sublner lgebr of V over the subset T = C( 3Z I ) S. Hvng seen exmples of substructures n cse of set specl neutrosophc complex lner lgebr over the set S, we now proceed onto gve exmples of semgroup complex neutrosophc lner lgebrs nd ther substructures. Exmple 1.57: Let 1 3 V = 5 6 C( Q I ); be semgroup lner lgebr of complex neutrosophc numbers over the semgroup S = Z + {}. Exmple 1.58: Let P = {ll 1 1 neutrosophc complex numbers from C ( Q I )} be semgroup neutrosophc complex lner lgebr of complex numbers over the semgroup S = 5Z. Clerly V = {ll 1 1 upper trngulr neutrosophc complex numbers wth entres from C ( Q I )} P s semgroup neutrosophc complex lner sublgebr of P over S. Tke W = {ll 1 1 dgonl neutrosophc complex mtrces wth entres from C ( Q I )} P; W s subsemgroup complex neutrosophc lner sublgebr of P over the subsemgroup T = 15Z, subsemgroup of 5Z = S. 53

55 Exmple 1.58: Let V = C( Q I ); be semgroup complex neutrosophc lner lgebr over the semgroup S = Z. Let M 1 = C( Q I ); V be semgroup complex neutrosophc lner sublgebr of V over S = Z. M = 1 1, C( Q I ) V, s semgroup complex neutrosophc lner sublgebr of V over S = Z. M 3 = 1 1, C( Q I ) V s semgroup complex neutrosophc lner sublgebr of V over S = Z. We see V = W 1 + W + W 3 where W W j = (), 1, j 3. Thus V s drect sum of semgroup lner sublgebrs of complex neutrosophc numbers over S = Z. 5

56 Let M 1 = M = M 3 = 1 C( Q I ); C( Q I ); C( Q I ); V, V, V, M = C( Q I );1 8 8 V, M 5 = 1 3 C( Q I ); V be semgroup complex neutrosophc lner sublgebrs of V over the semgroup S. We see V = 5 = 1 but W W j (); f j, 1, j 5. Thus V s the pseudo drect sum of semgroup complex neutrosophc lner sublgebrs of V over S. W 55

57 Exmple 1.6: Let A = C( Q I ); be semgroup neutrosophc complex lner lgebr over the semgroup S = 1Z. Tke M = C( Q I );1 13 A; s subsemgroup neutrosophc complex lner sublgebr of A over the subsemgroup Z = T of S = 1Z. We cn hve severl such subsemgroup complex neutrosophc lner sublgebrs of A over T S. Now we gve exmples of specl semgroup neutrosophc complex lner lgebrs nd ther substructures. Exmple 1.61: Let V = C( Q I ); be specl semgroup complex neutrosophc lner lgebr over the semgroup C ( Z I ). 56

58 Exmple 1.6: Let M = C( Q I ); be specl semgroup neutrosophc complex lner lgebr over the semgroup C ( Z I ). Exmple 1.63: Let V = C( Q I );1 16, be specl semgroup neutrosophc complex lner lgebr over the semgroup S = C ( Q I ). We see V s of fnte dmenson nd dmenson of V s 16 over S. Tke M = {set of ll upper trngulr mtrces wth entres from C ( Q I )} V; V s specl semgroup complex neutrosophc lner sublgebr of V over S. Consder H = 1 C( Q I );1 3 V, H s specl subsemgroup complex neutrosophc lner sublgebr of V over the subsemgroup T = C ( Z I ) C ( Q I ). 57

59 Tke W 1 = 1 3 1,, 3, C( Q I ) V, W 1 s specl semgroup of complex neutrosophc lner sublgebr of V over C ( Q I ). W = 1 C( Q I ),1 V s lso sublner lgebr of V. W 3 = C( Q I ),1 1 3 V s lso specl semgroup complex neutrosophc lner sublgebr of V over C ( Q I ). Let 1 W = 1, C( Q I ) V; be specl semgroup neutrosophc complex lner sublgebr of V over the semgroup S. 58

60 W 5 = 1, C( Q I ) 1 V be specl semgroup neutrosophc complex lner sublgebr of V over the semgroup S. Clerly W 1 + W + W 3 + W + W 5 V wth W W j = () ; j; 1, j 5, but stll V s not drect sum of subspces. On the other hnd suppose we dd the specl semgroup complex neutrosophc lner sublgebr. W 6 = 1, C( Q I ) 1 V, then V = W 1 + W + W 3 + W + W 5 + W 6 nd W W j = () f j, 1, j 6. Thus V s drect sum of subspces. Exmple 1.6: Let V = C( Q I ); be specl semgroup neutrosophc complex lner lgebr over the neutrosophc complex semgroup S = C ( Q I ). W 1 = 1 3 1,,3 C( Q I ) V; 59

61 W = 1 3 C( Q I );1 V, 1 3 W 3 = 5 C( Q I );1 5 V 1 nd W = 3 C( Q I );1 6 V 5 6 be the collecton of specl semgroup complex neutrosophc lner sublgebr of V. We see V = = 1 nd W W j (); f j, 1, j, thus V s only pseudo drect sum of subspces of V. We cn s n cse of semgroup vector spces defne the noton of lner trnsformtons of specl semgroup complex neutrosophc lner lgebrs only f both these lner lgebrs re defned over the sme complex neutrosophc semgroup. We cn lso defne lner opertor of specl semgroup complex neutrosophc lner lgebrs over the semgroup of complex neutrosophc numbers. urther the noton of bss nd dmenson cn lso be defned. Now we proceed onto defne the noton of group neutrosophc complex vector spce nd other relted concepts. DEINITION 1.16: Let V be set of complex neutrosophc numbers wth zero, whch s non empty. Let G be group under ddton. We cll V to be group neutrosophc complex number vector spce over G f the followng condtons re true. W 6

62 . or every v V nd g V gv nd vg re n V...v = for every v V, s the ddtve dentty of G. We gve exmples of them. Exmple 1.65: Let V = b b,,(, b,c,d),b,c,d C( Q I ) c d c d be group complex neutrosophc vector spce over the group G = Z. Exmple 1.66: Let C( Q I ), V = x,, = be group complex neutrosophc vector spce over the group G = 3Z. Exmple 1.67: Let V = C( Q I ), be group complex neutrosophc vector spce over the group G = Q. 61

63 Exmple 1.68: Let V = ( ) = 1 x,, 1,,..., 8 C( Q I ), 1 be group complex neutrosophc vector spce over the group G = Q. Consder C( Q I ), H = x,, (, 1,,,...) = 15 V; H s group complex neutrosophc vector subspce over the group G = Q. Tke 1 P = x,( 1,, 3,,,...,) C( Q I ), 1 = V; P s group complex neutrosophc vector subspce of V over G. Exmple 1.69: Let V = , 5 6, x 11 1 = C ( Q I ); } be group complex neutrosophc vector spce over the group G = Z. 6

64 Consder W 1 = C( Q I ),1 V, nd 1 3 W = 5 6 C( Q I ),1 9 V W 3 = x C( Q I ), 9 V = be group complex neutrosophc vector subspces of V over G. urther V = W 1 + W W 3 ; W W j (); f j, 1, j 3. Let B 1 =, x C( Q I ), 1... V = B = C( Q I ), V nd B 3 =, 5 6, x... 1 = C ( Q I ); 1 1} V be group complex neutrosophc vector subspces of V over the group G. Clerly V = B 1 B B 3 ; B B j φ; 1, j 3, hence V s pseudo drect unon of subspces. 63

65 We sy group complex neutrosophc vector spce V over group G s sd to be group complex neutrosophc lner lgebr over the group G f V s group under ddton. We gve exmples of group complex neutrosophc lner lgebr over group G. Exmple 1.7: Let V = 8 x C( Q I ), 8 = be group complex neutrosophc lner lgebr over the group G = Z. Exmple 1.71: Let V = C( Q I ),1 6 be group neutrosophc complex lner lgebr over the group G = 3Z. Exmple 1.7: Let V = C( Q I ),1 16 be group neutrosophc complex lner lgebr over the group G = Z. 1 3 W = C( Q I ),1 8 V;

66 be group neutrosophc complex lner sublgebr of V over the group G = Z. M = 1 3 C( Q I ), V be group neutrosophc complex neutrosophc complex lner sublgebr of V over the group G = Z. Exmple 1.73: Let V = C( Q I ), be group complex neutrosophc lner lgebr over the group G = Z. Tke P 1 = 1 3 C( Q I ),1 3 V, P = < > 8 1 C( Q I ),1 8 V, 65

67 P 3 = C( Q I ),1 6 V, nd P = C( Q I ), V P 5 = C( Q I ), V be group complex neutrosophc lner sublgebrs of V over the group G. Now we hve V = P 1 + P + P 3 + P + P 5 nd P P j = () f j, 1, j 5. Thus V s the drect sum of group complex neutrosophc vector subspces P 1, P,, P 5 of V over G. 66

68 Exmple 1.7: Let V = C( Q I ),1 1 be specl semgroup lner lgebr of neutrosophc complex numbers over the complex neutrosophc semgroup S = C( Z I ). Tke P = C( Z I ),1 1 V, P s specl semgroup lner sublgebr of complex neutrosophc numbers over the neutrosophc complex semgroup S = C ( Z I ). We see P cnnot be used to fnd drect sum of sublner lgebrs however, P cn be used n the pseudo drect unon of sublner lgebrs. urther we defne specl subsemgroup pseudo complex neutrosophc lner sublgebr of V. We sy proper subset T of V s pseudo specl semgroup complex neutrosophc lner sublgebr of V over the pseudo subsemgroup B of S f B s just semgroup of rels nd T s lso only rels. We proceed onto gve exmples of ths stuton. Exmple 1.75: Let 1 3 V = C( Z I ), be specl semgroup complex neutrosophc lner lgebr over the complex neutrosophc semgroup S = C ( Z I ). 67

69 Let P = Q,1 6 V, P s not specl semgroup lner sublgebr over the semgroup S = C ( Z I ), but P s semgroup lner lgebr over the semgroup Z = T. Then P s defned / clled s the pseudo specl semgroup of complex neutrosophc lner sublgebr over the pseudo subsemgroup T = Z. Infct P s lso pseudo specl semgroup complex neutrosophc lner sublgebr over the pseudo specl subsemgroup M = 3Z + {}. Thus we hve nfnte number of pseudo specl semgroup complex neutrosophc lner sublgebrs over the pseudo specl subsemgroups N of S. Now we cn lso hve pseudo specl neutrosophc semgroup lner sublgebrs over the pseudo neutrosophc subsemgroup B of S. We wll llustrte ths stuton lso by exmples. Exmple 1.76: Let V = 1 3 C( Q I ),1 be specl complex neutrosophc lner lgebr over the complex neutrosophc semgroup S = C ( Z I )}. Tke 1 W = ( Q I ),1 3 V be the pseudo neutrosophc subsemgroup specl complex neutrosophc lner sublgebr of V over the pseudo neutrosophc subsemgroup B = Z I of S. 68

70 Now consder A = 1 3 C( Z I ),1 be pseudo specl subsemgroup complex neutrosophc lner sublgebr of V over the pseudo neutrosophc subsemgroup 3Z I of the neutrosophc complex semgroup S. Consder 1 N = ( Z I ),1 3 V, N s pseudo specl subsemgroup lner sublgebr of neutrosophc complex numbers of V over the neutrosophc subsemgroup T = {3Z I} S. We cn hve severl such exmples. The followng theorem s suffcent to prove these. THEOREM 1.11: Let V be specl semgroup neutrosophc complex lner lgebr over the complex neutrosophc semgroup S. 1. V hs pseudo specl neutrosophc subsemgroup complex neutrosophc lner sublgebrs over the pseudo neutrosophc subsemgroup of S.. V hs pseudo specl ordnry subsemgroup complex neutrosophc lner sublgebr over the pseudo rel subsemgroup of S. Exmple 1.77: Let 1 3 V = C( Q I ), be specl semgroup complex neutrosophc lner lgebr over the semgroup S = C ( Q I ). Clerly dmenson of V 69

71 7 over S s 16; however f S s replced by T = C ( Z I ) S then the dmenson of V over T s nfnte. So we cn hve specl subsemgroup lner lgebrs to be of nfnte dmenson even when the dmenson of the specl semgroup lner lgebr s fnte dmenson over the semgroup S but of nfnte dmenson over the subsemgroup of S. Exmple 1.78: Let V = C( Q I ) be specl semgroup lner lgebr of complex neutrosophc number over C ( Q I ). Dmenson of V s one. Clerly V hs no specl semgroup lner sublgebrs but V hs pseudo ordnry specl subsemgroup lner sublgebrs nd pseudo neutrosophc specl subsemgroup lner sublgebrs. or M = C( Z I ) V, s pseudo neutrosophc subsemgroup lner sublgebr over the pseudo neutrosophc subsemgroup T = Z I C( Q I ). Lkewse

72 N = Z s pseudo ordnry subsemgroup neutrosophc complex lner sublgebr over the pseudo ordnry subsemgroup B = Z S. Thus we cn sy, V s specl semgroup neutrosophc complex lner lgebr over S to be smple f t hs no proper specl semgroup neutrosophc complex lner sublgebr over S. Exmple 1.79: Let V = C( Q I ) be specl semgroup neutrosophc complex lner lgebr over the semgroup S = C ( Q I ). V s smple. However V hs specl subsemgroup complex neutrosophc lner sublgebrs M over the subsemgroup T = C( Z I ) C( Q I ) = S where M = C( Z I ) V. Also we consder b b b N = b b b C( 3Z I ) V; b b b N s specl subsemgroup neutrosophc complex lner sublgebr of V over the subsemgroup B = {C ( Z I )} S. 71

73 DEINITION 1.17: Let V be ordnry (neutrosophc or specl) semgroup complex neutrosophc lner lgebr over the ordnry semgroup S (or neutrosophc semgroup or complex neutrosophc semgroup). If on V we cn defne product nd V s comptble wth product (V s semgroup wth respect to nother operton prt from ddton) then we defne V to be ordnry (neutrosophc or specl) semgroup double lner lgebr over the ordnry semgroup S (or neutrosophc semgroup or neutrosophc complex semgroup). We wll llustrte ths stuton by some smple exmples. Exmple 1.8: Let 1 3 V = 5 6 C( Q I ), be ordnry semgroup complex neutrosophc lner lgebr over S = Z. V s clerly ordnry semgroup complex neutrosophc double lner lgebr over S where S = Z. Exmple 1.81: Let P = x C( Q I ) = be ordnry semgroup complex neutrosophc double lner lgebr over the semgroup S = 3Z + {}. We cn defne substructure, bss, lner opertor nd lner trnsformton whch s just mtter of routne. We hve the followng nterestng result. THEOREM 1.1: Every ordnry semgroup complex neutrosophc double lner lgebr over the semgroup S s ordnry semgroup complex neutrosophc lner lgebr over S but however n generl ordnry semgroup complex neutrosophc lner lgebr s not double lner lgebr. 7

74 or the ltter prt of the proof we gve n exmple. Exmple 1.8: Let M = C( Q I ); be n ordnry semgroup complex neutrosophc lner lgebr over the semgroup S = Z + {}. Clerly M s not n ordnry semgroup complex neutrosophc double lner lgebr over S. Exmple 1.83: Let V = x C( Q I ) = be n ordnry semgroup complex neutrosophc double lner lgebr over the semgroup S = 3Z + {}. Consder M = x C( Z I ) V; = M s lso n ordnry semgroup complex neutrosophc double lner sublgebr of V over the semgroup S. Tke P = x C( Q I ) V; = P s only pseudo ordnry semgroup complex neutrosophc double lner sublgebr of V over S or ordnry semgroup complex neutrosophc pseudo double lner sublgebr of V over S for on P product cnnot be defned. Ths sme concept of double lner lgebr cn be esly extended to the cse of specl semgroup complex neutrosophc 73

75 lner lgebrs, complex semgroup complex neutrosophc lner lgebrs nd neutrosophc semgroup complex neutrosophc lner lgebrs. The defnton s mtter of routne so we gve only exmples of them. Exmple 1.8: Let M = x C( Q I ) = be neutrosophc semgroup neutrosophc complex double lner lgebr over the neutrosophc semgroup S = Z I. Tke 1 P = x C( Z I ) M; = P s only neutrosophc semgroup neutrosophc complex lner sublgebr whch s not double lner sublgebr of M. We cll T neutrosophc semgroup neutrosophc complex pseudo double lner sublgebr of M. Thus prt from double lner sublgebrs we cn lso hve pseudo double lner sublgebrs of M. Exmple 1.85: Let V = C( Q I ),1 1 be complex semgroup neutrosophc complex semgroup neutrosophc complex lner lgebr over the complex semgroup C (Z) = { + b, b Z} = S. Clerly V s not double lner lgebr. Exmple 1.86: Let S = 1 3 C( Q I ),1 7

76 be complex semgroup complex neutrosophc double lner lgebr over the complex semgroup T = C (Q) = { + b, b Q}. We see X =,b C( Q I ) b S s only complex semgroup complex neutrosophc pseudo double lner sublgebr of S over T. or we see f c A = b nd B = d n X then c d AB = b d = bc X. Hence X s only pseudo lner sublgebr of S over T. Let b V =,b,c C( Q I ) c S, be gn complex semgroup neutrosophc complex double lner sublgebr of S over the semgroup T. Let us consder L =,b C( Q I ) b S, L s gn complex subsemgroup neutrosophc complex double lner sublgebr over the complex subsemgroup C (Z) = { + b, b Z} C(Q) = T. Now hvng seen neutrosophc nd complex double lner lgebr we now proceed onto gve exmples of specl semgroup neutrosophc complex double lner lgebrs. 75

77 Exmple 1.87: Let V = C( Q I ), be specl complex neutrosophc double lner lgebr over the neutrosophc complex semgroup S = {C ( Q I )}. Consder A = C( Q I ),1 6 6 V, A s specl complex neutrosophc double lner sublgebr of V over the complex semgroup S = {C ( Q I )}. Tke B = 1 C( Q I ),1 3 3 V; B s specl neutrosophc complex pseudo lner sublgebr of V over S. or f x = 1 3 nd y = b1 b b3 B; 1 xy = 3 b1 b b3 76

78 = 1b3 b 1b 3 B so B s only pseudo lner sublgebr of V over S. Now consder L = 1 3 C( Q I ), V, be specl subsemgroup complex neutrosophc double lner sublgebr of V over the specl subsemgroup T = C ( Z I ) S; of S. Let 1 3 P = 5 6 C( Z I ),1 9 V be pseudo specl complex subsemgroup complex neutrosophc double lner sublgebr of V over the complex subsemgroup N = {( + b), b Z} S, N pseudo specl subsemgroup of S. Let 1 3 C = 5 6 ( Z I ),1 9 V be pseudo neutrosophc specl subsemgroup of double lner sublgebr of V over the neutrosophc subsemgroup E = { Z I } S, E s pseudo neutrosophc specl subsemgroup of V over S. We cn defne bss, specl double lner trnsformton, double opertor nd so on s n cse of usul semgroup lner 77

79 lgebrs n cse of semgroup neutrosophc complex lner lgebrs. Ths tsk s lso left s n exercse to the reder. Now we wll proceed onto contnue to defne nd work to group neutrosophc complex vector spces / lner lgebrs. Exmple 1.88: Let V = C( Q I ), be group complex neutrosophc lner lgebr over the group S = Z. Let 1 3 M = C( Z I ),1 1 V be pseudo semgroup complex neutrosophc lner sublgebr of V over the semgroup T = Z + {} Z. Tht s T s Smrndche specl defnte group s t hs proper set whch s semgroup. We defne group neutrosophc complex lner lgebr V to be group neutrosophc complex double lner lgebr f V s endowed wth nother operton product. We wll llustrte ths stuton by some exmples. Exmple 1.89: Let M = C( Q I ),

80 be group neutrosophc complex double lner lgebr over the group G = Q. We cn fnd substructures of M. Clerly M s lso comptble wth respect to mtrx multplcton. Exmple 1.9: Let V = C( Q I ),1 16 be group neutrosophc complex double lner lgebr over the group G = Q. Consder M = C( Z I ),1 16 V, M s subgroup neutrosophc complex double lner sublgebr of V over the subgroup H = Z Q = G. Tke N = 1 C( Q I ),1 3 V, be group neutrosophc complex double lner sublgebr of V over the group G = Q. 79

81 Let L = 1 3 C( Q I ),1 V, L s lso group neutrosophc complex double lner sublgebr of V over the group G = Q. Let A = 1 3 C( Q I ),1 V be group complex neutrosophc lner sublgebr of V, nfct double lner sublgebr of V over G. or f then x = b c d nd y = e f g h A b c d xy = e f g h =. Thus A s group complex neutrosophc double lner sublgebr of V over G. 8

82 Let C = 1 C( Q I ),1 3 V be group neutrosophc complex pseudo double lner sublgebr of V over the group G. or f 1 b1 x = b nd y = 3 b3 b re n C. Now consder the product 1 b1 x.y = b 3 b3 b = 1b b3 3b b1 C. So C V s not double lner sublgebr, hence C s only group complex neutrosophc pseudo double lner sublgebr of V over G. Now hvng seen exmples of them we cn proceed on to defne bss, lner opertor, lner trnsformton, drect sum, pseudo drect sum s n cse of semgroup neutrosophc complex double lner lgebrs. Ths tsk s left s n exercse to the reder. 81

83 We cn defne fuzzy neutrosophc complex groups, semgroups etc n two wys. or f we wnt to nduct complex fuzzy number we see 1 = so f product s to be defned we need to nduct = 1 but f we wsh to work only wth mn mx then we do n the followng wy. Let C ( [,1] [,I] ) = { + b + ci + di where, b, c, d [, 1]}, we defne C ( [,1] [,I] ) to be the fuzzy complex neutrosophc numbers. We defne mx or mn operton on C ( [,1] [,I] ) s follows: If x = + b + ci + di nd y = m + n + ti + si re n C ( [,1] [,I] ); then mn (x, y) = mn (+b+ci+di, m+n+ti+si) = mn (, m) + mn (b, n) + mn (ci, ti) + mn(di, si). It s esly verfed mn (x, y) s gn n C ( [,1] [,I] ). Consder x = I + (.8)I nd y = I + (.7)I n C ( [,1] [,I] ). Now mn {x, y} = mn { I + (.8)I, I + I (.7)I} = mn {.7,.9} + mn {.61,.3} + mn {.3I,.193I} + mn {(.8)I, I (.7)I} = I + (.8)I. Thus {C ( [,1] [,I] ), mn} s semgroup. Lkewse we cn defne the operton of mx on C ( [,1] [,I] ) nd {C ( [,1] [,I] ) mx} s lso semgroup. These semgroups wll be known s fuzzy neutrosophc complex semgroup. Now we defne specl fuzzy complex neutrosophc group, semgroup nd rng s follows: DEINITION 1.18: Let V be ordnry semgroup neutrosophc complex vector spce over the semgroup S. Let η be mp from V nto C ( [,1] [,I] ) such tht (V, η) s fuzzy semgroup neutrosophc complex vector spce or semgroup neutrosophc complex fuzzy vector spce [ ]. 8

84 Lkewse one cn defne for neutrosophc semgroup neutrosophc complex vector spce, complex semgroup neutrosophc complex vector spce, specl semgroup neutrosophc complex vector spce over the neutrosophc semgroup, complex semgroup nd complex neutrosophc semgroup respectvely the fuzzy nlogue. urther we cn defne for set complex neutrosophc vector spce nd group complex neutrosophc vector spce over set nd group respectvely lso the fuzzy nlogue. Thus defnng these fuzzy notons s mtter of routne nd hence left s n exercse to the reder. We gve exmples of these stutons. Exmple 1.91: Let V = C( Z I ), be semgroup complex neutrosophc lner lgebr over the semgroup S = Z. Let η : V C ( [,1] [,I] ) 1 f Z f C(Z) 1 3 η( 5 6 ) = I 7 8 f Z I 9 I f C( Z I ) If = then 1. If every = then 83

85 η = 1. where A = η (A + B) mn (η (x), η (y)) 1 3 b1 b b3 5 6 nd B = b b5 b b7 b8 b 9 for every A, B V nd η (rx) η (x), r Z. Exmple 1.9: Let V = 1 x,,(,,..., ) C( Q I ), = 3 be set complex neutrosophc vector spce over the set S = 3Z Z 5Z. Defne η : V C ( [,1] [,I] ) 1 f 1 f = f s complex η ( ) = I f I s neutrosophc I f I. 8

86 wth ths stpulton (V, η) s set fuzzy complex neutrosophc vector spce over Z. Our mn crter s to bult the mp η such tht η () =, η (I) = I nd η (I) = I wth ths ordnry mppng η : V C ( [,1] [,I] ) we get the fuzzy complex neutrosophc structures. Now other wy of defnng fuzzy structures of complex neutrosophc elements s s follows. DEINITION 1.19 : Let V = {(x 1,, x n ) x C ( [, 1] [, I] ), 1 n}; V s fuzzy complex neutrosophc semgroup wth mn (or mx) functon. or used n the mutully exclusve sense. Exmple 1.93: Let V = {(x 1, x, x 3 ) x C ( [,1] [,I] ); 1 3}, V s fuzzy complex neutrosophc semgroup under mx. Exmple 1.9: Let x1 x V = x1 x C ( [,1] [,I] ); 1 1} be fuzzy complex neutrosophc semgroup under mn functon. Exmple 1.95: Let M = {ll 1 3 fuzzy neutrosophc complex mtrces wth entres from C ( [,1] [,I] )}. M under mx (or mn) s semgroup. or f 1 3 b1 b b3 5 6 X = b b5 b nd Y = b8 b9 b3 be n M, 85

87 mn (X, Y) = mn 1 3 b b b 5 6, b b b b b b = mn{ 1,b 1} mn{,b } mn{ 3,b 3} mn{,b } mn{ 5,b 5} mn{ 6,b 6}. mn{ 8,b 8} mn{ 9,b 9} mn{ 3, b 3} (If mn s replced by mx stll we get semgroup). Exmple 1.96: Let V = C( [,1] [,I] ); V s semgroup under mx (or mn). Now we cn defne only set semvector spce over the set S. DEINITION 1.: Let V be set of fuzzy complex neutrosophc elements. S = {, 1} be set; V s set semvector spce over the set S f ) or v V nd s S we hve vs = sv V. ).v = V. 1.v = v V, we defne V to be fuzzy neutrosophc complex semvector spce over the set S. We wll gve exmples of set fuzzy neutrosophc complex semvector spce over the set S. 86

88 Exmple 1.97: Let V = {(x 1, x, x 3, x ), x1 x1 x x 3 x 6 x x5 x 6, x 3, x x, j C ( [,1] [,I] ), = x7 x8 x 9 x x 5 1 9; j 6} be set of fuzzy neutrosophc complex elements. V s set fuzzy neutrosophc complex semvector spce over the set S = {,1}. Exmple 1.98: Let V = {(x 1, x,, x 9 ), x1 x 1 3 x3 x , x x x, j C ( [,1] [,I] ), 1 ; 1 j 16} be set fuzzy complex neutrosophc semvector spce over the set S ={, 1}. Exmple 1.99: Let V = where C ( [,1] [,I] ), 1 6} be semgroup under mx operton V s set fuzzy complex neutrosophc semlner lgebr over the set S = {, 1}. 87

89 Exmple 1.1: Let M = C( [,1] [,I] ), 1 8} be set fuzzy neutrosophc complex sem lner lgebr over the set X = {, 1}. Now we cn defne substructures, bss, lner trnsformton nd lner opertor for these set fuzzy complex neutrosophc semvector spces / semlner lgebrs over the set S. Ths tsk s left s n exercse to the reder. DEINITION 1.1: Let V be set wth elements from C ( [,1] [,I] ) nd S be semgroup wth mn or mx or product. We defne V to be semgroup fuzzy neutrosophc complex semvector spce over S f the followng condtons hold good; ) s.v V for ll v V nd s S. ).v = V for ll v V nd S. Exmple 1.11: Let 1 1 C( [,1] [,I] ); V =,( 1,,..., 9), 3 1 be semgroup fuzzy complex neutrosophc semvector spce over the semgroup S = {, 1} under multplcton. Results n ths drecton cn be got wthout ny dffculty. Thus one cn defne structures on C ( [,1] [,I] ) but the scope s lmted. However f we consder mp η : V C ( [,1] [,I] ) we cn hve lmost defne ll lgebrc fuzzy neutrosophc complex structures wthout ny dffculty. 88

90 Chpter Two INITE COMPLEX NUMBERS We know s complex number where = 1 or 1 = where 1 s from the set of rels. However here we defne fnte complex numbers n modulo ntegers whch re n fnte set up. Through out ths book Z n wll denote the set of modulo ntegers {, 1,,, n 1} nd 1 s (n 1), = (n ) nd so on. Thus we use the fct 1 = n 1 to defne the fnte complex modulo numbers. DEINITION.1: Let C (Z n ) = { + b, b Z n, s the fnte complex modulo number such tht = n 1, n < } we defne s the fnte complex modulo number. C (Z n ) s the fnte complex modulo nteger numbers. It s nterestng to note tht snce fnte vlues n Z n re dependent on n so lso the fnte complex number s lso dependent on Z n for every n. We gve exmples of them. Exmple.1: Let C (Z ) = {, 1,, 1 + }. We see = 1 = 1 = 1. Also ( + 1) = = 1+1 =. C(Z ) s rng. 89

91 Exmple.: Let C (Z 3 ) = { + b, b Z 3 } = {, 1,,,, 1 +, +, + = 1 = 3 1 = } ( ) = = 1 = ( + 1) = = = (mod 3). (1 + ) = 1 + ( ) + (mod 3) = (mod 3) = (mod 3). ( + ) = + ( ) + (mod 3) = + + = (mod 3). ( + ) = ( ) (mod 3) = (mod 3) = (mod 3). We gve the tbles ssocted wth C(Z ) nd C(Z 3 ). Multplcton tble C(Z ). X Tble for C(Z 3 ) X We see C(Z ) s rng wth zero dvsor of chrcterstc two where s C(Z 3 ) s feld of chrcterstc three nd o(c(z 3 )) = 9. 9

92 Exmple.3: Let C(Z ) = { + b, b Z } = {, 1,, 3,,, 3, 1+, +, 3+, 1+, 1+3, +, +3, 3 +, 3 +3} s complex rng of order 16. Exmple.: C(Z 5 ) = { + b, b Z 5 } = {, 1,, 3,,,, 3,,, + } s only fnte complex rng for (1 + ) ( + ) = s zero dvsor n C(Z 5 ). Exmple.5: Consder C(Z 7 ) = { + b, b Z 7 } be the fnte complex rng C(Z 7 ) s feld. or tke ( + b ) (c + d ) = where + b, c + d C(Z 7 ) wth, b, c, d Z 7 \ {}. ( + b ) (c + d ) = mples c + 6bd = (1) d + bc = () (1) + () b gves c + 6bd = b c + bd = c ( + b ) = c ths forces + b = n Z 7. But for no nd b n Z 7 \ () we hve + b = so C(Z 7 ) s fnte complex feld of chrcterstc seven. We see C(Z 11 ) s gn fnte complex feld of chrcterstc eleven C(Z 13 ) s not fnte complex feld only rng for 9 + nd + 9 n C (Z 13 ) s such tht (9 + ) (3 + 9 ) =. In vew of ll these we hve the followng theorem whch gurntees when C (Z p ) s not feld. THEOREM.1: Let C(Z p ) be the fnte complex number rng. C(Z p ) s not feld f nd only f there exsts, b Z p \ {} wth + b p or + b (mod p) where p s prme. Proof: Let C(Z p ) = { + b, b Z p, = p 1}. To show C(Z p ) s not feld t s enough f we show C(Z p ) hs zero dvsors. Suppose C(Z p ) hs zero dvsors sy + b nd c + d n C(Z p ) (, b, c, d Z p \ {}) s such tht ( + b ) (c+d ) =, 91

93 then c + bd + (d + bc) = tht s c + (p 1) bd + (d + bc) = ths forces c + (p 1)bd = () nd bc + d = () () + () b gves c + (p 1) bd = + b c + bd = (snce, b, c, d re n Z p \ {} nd p s prme, no tem s zero) gves c + b c = tht s c ( + b ) = s c Z p \ {} nd c -1 exsts s p s prme. We see c ( + b ) = s possble only f + b. Conversely f + b = thn we hve ( + b ) (b+ ) = for consder, ( + b ) (b + ) = b + (p 1)b + ( +b ) pb + ( +b ) (mod p) s, b Z p nd gven + b =. Thus C (Z p ) hs zero dvsors hence C (Z p ) s rng nd not feld. THEOREM.: Let C(Z p ) be the commuttve fnte complex rng, p prme, C(Z p ) s feld f nd only f Z p hs no two dstnct elements nd b dfferent from zero such tht + b (mod p). Proof: ollows from the fct tht f C(Z p ) hs no zero dvsors t s commuttve ntegrl domn whch s fnte hence s feld by [ ]. Now we wll derve other propertes relted wth these fnte complex rngs / felds. Let C(Z) = { + b, b Z} be the collecton of complex numbers. Clerly C(Z) s rng. Consder the del generted by +, denote t by I. C(Z) = {I, 1 + I, + I, 1+ + I} I where (1 + ) + I = + + I = I s here 1 (mod ) nd = mod. 9

94 Now we denote t by nd C(Z) C(Z ). Consder 3 + 3I I C(Z). Let J be the del generted by 3 + 3I. C(Z) J = 3 + 3I = {J, 1 + J, + J, + J, + J, I, + + J, J, + + J}. Here lso + 1 (mod 3) nd + 1 (mod 3) further = 1 =. We use for for one cn understnd from the fct = n 1. So we cn sy C(Z) C (Z 3 ) J where =. Thus we get relton between C(Z) C(Z n ) nd. J = n + n We hve n C(Z n ) zero dvsors, dempotents, nlpotent everythng dependng on n. We cll C(Z p ) when C (Z p ) s feld s the complex Glos feld to honour Glos. However for every prme p, C(Z p ) need not n generl be complex Glos feld. Here we gve some propertes bout C(Z p ), p prme or otherwse. Consder x = + b C(Z p ) then x = + (n 1) b C(Z p ) s defned s the conjugte of x nd vce vers. We see x. x = ( + b) ( + (n 1) b) = + b + (n 1) b + (n 1)b = + + (n 1) b = + b. Recll f + b (mod n) then C(Z n ) hs nontrvl zero dvsor. We cn dd nd multply fnte complex number usng = (n 1) f elements re from C(Z n ). 93

95 Thus f x = + b nd y = c + d re n C(Z n ) then x + y = ( + b) + (c + d) = ( + c) modn + (b + d) mod n. Lkewse x.y = ( + b) (c + d) = c + bc + d + bd = c + bc + d + (n 1)bd = (c + (n 1)bd) (mod n) + (bc + d) (mod n). The operton + nd re commuttve nd ssoctve. It s mportnt to note they re modulo n ntegers nd they re not orderble (n ) we cn hve for + b = x, x -1 exst or need not exst n C(Z n ). or n C(Z ) we see x = 1 + C(Z ) nd x -1 does not exst s (1 + ) = (mod ) Consder x = 3 + n C(Z 7 ) we hve y = n C(Z 7 ) such tht xy = 1. Consder xy = (3+ ) (6+ 6) = = = = = 1 (mod 7) s 7 = mod 7 nd 8 1 mod 7. Other propertes of usul complex numbers re not true n cse of fnte complex modulo numbers. We cn gve grphcl representton of complex modulo ntegers n 3 lyers. The nner most lyer conssts of rel modulo ntegers mkes s n the fgure. The outer lyer conssts of complex modulo numbers, where s the outer most lyer s the mxed complex modulo nteger we represent the grph or dgrm for C (Z )

96 The dgrm for C (Z 3 ) The dgrm for C (Z ) o(c(z )) = 16 nd s rng we cn drw for ny C(Z n ); n. We cnnot do ny plne geometry usng these fnte complex modulo ntegers but we cn get severl lgebrc results on C(Z n ); n. Our mn motvton to ntroduce these concepts s to ntroduce fnte neutrosophc complex modulo number / ntegers. Just before we proceed to defne these concepts, we defne substructures n C(Z n ); n. DEINITION.: Let C (Z n ) be the rng of fnte complex modulo nteger / numbers. Let H C(Z n ) (n ) f H tself s rng under the opertons of C(Z n ). We cll H subrng of fnte complex modulo ntegers C(Z n ). We gve some exmples of them

97 Exmple.6: Let C(Z ) be the rng of fnte complex modulo ntegers C(Z ) = { + b, b Z } P = {, 1+ = 1} C(Z ) s subrng of C(Z ). Exmple.7: Let C(Z 3 ) = { + b, b Z 3 }, be the rng. P = {1,, } C(Z 3 ) s subrng of C(Z 3 ; P = Z 3 ) s the rng of modulo ntegers three. We see =, so we cnnot fnd subrng n C(Z 3 ). It s pertnent to menton here tht for Z s zero Z 3 s such tht = 1, Z s such tht =, Z 5 s such tht 3 = 1 (mod 7) nd so on. Lkewse s fnte, = 1 (mod ), ( ) = (mod 3), ( ) = 3(mod ) nd so on ( ) = (n 1) (mod n). Exmple.8: Let C(Z 6 ) = { + b, b Z 6, = 5}, C(Z 6 ); (3 + 3 ) = = = = = (mod 6). or 1 + C(Z 6 ) we hve (1+ ) = = =. We hve n nterestng theorem. THEOREM.3: Let C (Z p ) be the fnte complex rng, p prme p >. C (Z p ) s rng nd (1+ ) =. Proof: Consder (1+ ) = 1+ + = p 1 = s p =. Now to show C (Z p ) s rng t s enough f we prove the exstence of zero dvsor. Tke p + p n C (Z p ), (p + p) = p (1+ ) = p ( ) = p. = p (p ) (mod p). Thus C (Z p ) hs zero dvsors, hence C (Z p ) s rng. 96

98 COROLLARY.1: Let C (Z n ) be ny complex fnte rng feld. If x = 1+ C (Z n ) then x =. Proof: Consder 1 + = x C (Z n ), now (1 + ) = + + = 1 + n 1 + = (mod n). Hence the clm. It s nfct dffcult tsk to fnd subrngs nd dels n C(Z n ) where o(c(z n )) = n. Exmple.9: Let C(Z ) be the fnte complex rng. I = {, + } C(Z ) s n del of C(Z ). P = {, 1,, 3} C(Z ) s only subrng of C (Z ) nd not n del. We see S = {,,, 3 } C (Z ) S s not even subrng only set under multplcton s = 3. So n generl we cnnot get ny nce lgebrc structure usng {,,, 3 } = S; S cn only be group under ddton nd not even semgroup under multplcton. THEOREM.: Let C (Z n ) be the rng, Z n C (Z n ) s subrng of C (Z n ) nd not n del of C (Z n ). Proof s drect nd hence left s n exercse to the reder. THEOREM.5: Let C (Z p ) be the rng. P = {, p + p } C (Z p ) s the del of C (Z p ), p prme. Proof: Let C (Z p ) be the gven rng, p prme. Consder S = {, p + p } C (Z p ). Clerly S under ddton s n beln group. Now (p + p ) ( + b ) = p + p + pb + pb ( ) = p + (p + pb) + (p 1) pb = p ( + (p 1)b) + p ( + b). = p + p (mod p) (usng smple number theoretc technques). Exmple.1: Let C(Z 6 ) = R be complex rng of modulo ntegers. C(Z 6 ) = { + b, b Z 6 }. Consder P = {,

99 13 } R; to show P s n del of R. (P, +) s n ddtve beln group. To show (P, ) s semgroup ( ). =, ( ) ( ) = ( = 5) = (5) = 13 (1+5) = 13.6 (mod 6) =. Thus (P, +, ) s rng. Consder ( + b ) P;, b Z 6 ; f = 3 nd b = 5 both nd b odd n Z 6, then (3 + 5 ) (13+13 ) = (mod 6) = (mod 6) = (mod 6) = + (mod 6) P. Now tke = 5 odd nd b = 8 even. + b = n R (5 + 8 ) ( ) = (mod 6) = (mod 6) = (mod 6) Hence s n P. Hence the clm. Exmple.11: Let R = C (Z 11 ) be complex modulo nteger rng wth = 1. Consder n R. (5+5 ) = (mod 11) = = 6. It s nterestng to see ( + ) s lwys b only n mgnry or complex modulo number further (6 + 6 ) = 6. We hve nce nterestng number theoretc result. THEOREM.6: Let R = C(Z p ), p prme be the rng of complex modulo ntegers = p 1, then () ( + ) = b, b, Z p nd p + 1 p + 1 ( p + 1) ( p + 1) () + = =. The proof uses only smple number theoretc technques hence left s n exercse to the reder. 98

100 Exmple.1: Let S = C(Z ) = { + b, b Z } be complex rng of modulo ntegers. Consder x = S, x = (8 + 1 ) = (mod ) = (mod ) = 1 (mod ). Thus f x = + ( ) C (Z ); then x = ( + ( ) ) = + ( ) 19 + ( ) (mod ) = ( ) (mod ) = (1 + 19) + c (mod ) = + c (mod ); c Z. Invew of ths exmple we hve n nterestng result. THEOREM.7: Let S = C (Z n ) be complex rng of modulo ntegers x = + (n ) C (Z n ) then x = b for some b Z n, = n 1. Smple number theoretc method yelds the soluton. Exmple.13: Let C (Z 1 ) = S be modulo nteger rng. Consder x = (6 + 6 ) n S, x = s zero dvsor n S. Tke y = S, y = = (mod 1) = 6. = +8 n S. = = n S. If we consder b = +6 n S, b = (+6 ) = = for.6 Z 1 s such tht.6 =. If n = 3+8 then n = (3 + 8 ) = = 5. Thus we see every x = + b n S for whch.b = mod 1 s such tht x Z 1 s rel vlue. THEOREM.8: Let R = C (Z n ) be rng of complex modulo ntegers (n not prme). Every x = + b n whch.b = (mod n) gves x to be rel vlue. Proof follows from smple number theoretc technques. We see ths rng S = C(Z n ) hs zero dvsors, unts nd dempotents 99

101 when n s not prme. Now we cn use these rngs nd buld polynoml rngs nd mtrx rngs. We wll gve exmples of them. Let S = C(Z n ) be complex modulo nteger rng. S[x] = [C(Z n )][x] s polynoml rng wth complex modulo nteger coeffcents. or we see f S[x] = C(Z 3 )[x] = x = where = m + n ; m, n Z 3 } we see S[x] s commuttve rng of nfnte order but of chrcterstc three. If p(x) = ( + ) + (1 + ) x + ( + )x 7 nd q(x) = + ( + )x 3 + x 6 re n S [x], p(x) q(x) = [( + ) + (1 + )x + ( + ) x 7 ] [ + ( + )x 3 + x 6 ] = ( + ) + ( + 1) x + ( + ) x 7 + ( + ) x 3 + (1+ ) (+ )x + (+ ) (+ ) x 1 + (+ )x 13 + (+ )x 6 + (1+ ) x 7 = ( +) + (1 + )x +(1 + )x 7 + (1 + + )x 3 + ( )x + ( )x 1 + (1 + )x 13 + (1 + )x 6 + ( + )x 7 = ( + ) + (1+ )x + x 3 + x + x 1 + (1 + )x 13 + (1 + )x 6 + ( + )x 7. Ths s the wy the product of two polynomls s performed. 9 Exmple.1: Let S [x] = C (Z 8 ) [x] = x C (Z 8 ) = = {x + y x, y Z 8 }; 9} be the set of polynomls n the vrble x wth coeffcents from C (Z 8 ). S [x] s n ddtve beln group of complex modulo nteger polynomls of fnte order. Clerly S [x] s not closed wth respect to product. 9 Clerly S[x] contns Z 8 [x] = x Z8 S [x], the = usul polynoml group under ddton of polynomls. 9 Z 8 [x] = x such tht = x wth x Z 8 nd = 7} = 1

102 S[x] s pure complex modulo nteger group of fnte order contned n S[x] under ddton. Thus S[x] = Z 8 [x] + Z 8 [x] wth Z 8 [x] Z 8 [x] = {} s the drect sum of subgroups. Other thn these subgroups S[x] hs severl subgroups. Exmple.15: Let 3 S[x] = C[Z ] [x] = x = x + y ;x, y Z = = {, 1,,,, x, x, x, x, x 3, x 3, x 3 nd so on} be complex beln group under ddton of modulo complex ntegers of fnte order. Exmple.16: Let S[x] = x C (Z 8 ) = { + b, b Z 8, = = 79}} be the set of polynomls n the vrble x wth complex modulo nteger coeffcents. S[x] s semgroup under multplcton. We hve the followng nterestng theorems; the proof of whch s smple nd hence s left s exercse to the reder. THEOREM.9: Let S [x] = n x n <, C (Z n ) = { + = b = n 1 nd, b Z n }. S [x] s group of complex modulo nteger polynomls under ddton. THEOREM.1: Let S [x] = x C (Z n ) = { + b =, b Z n nd = n 1}. (1) S [x] s group under ddton of nfnte order. () S [x] s commuttve monod under multplcton. Exmple.17: Let S[x] = C(Z 18 )[x] = x C(Z 18 ) = {x = + y x,y Z 18 nd =17} be the collecton of ll polynomls. S[x] s rng. S[x] hs zero dvsors, unts nd dempotents. 11

103 It s nterestng to note tht these polynoml rngs wth complex modulo ntegers cn be studed s n cse of usul polynomls lke the concept of reducblty, rreducblty fndng roots etc. Ths cn be treted s mtter of routne nd cn be done wth smple nd pproprte chnges. Now we proceed onto study nd ntroduce the noton of mtrx rng wth complex modulo nteger entres. We wll llustrte ths by some exmples. Exmple.18: Let V = {(x 1, x,, x 9 ) x C(Z 7 ) = { + b, b Z 7 nd = 6}; 1 9} be the 1 9 row mtrx (vector). Clerly V s group under ddton of fnte order nd V s semgroup under product of fnte order. V s semgroup hs zero dvsors. Ths semgroup (V, ) s Smrndche semgroup. Exmple.19: Let P = {(x 1, x, x 3, x ) x C (Z 1 ) = { + b, b Z 1 nd = 11, 1 } be the collecton of 1 row mtrces wth complex modulo nteger entres. P s group under ddton of fnte order nd (P, ) s semgroup of fnte order hvng dels, subsemgroups, zero dvsors, nd unts. Now we show how ddton s crred out. Let x = (7 + 3, +,, 7) nd y = (, 5, 9 +, 1 + ) be n (P, +). x + y = (7 + 3, +,, 7) + (, 5, 9 +, 1 + ) = ( , + + 5, ) (mod 1) = (9 + 3, + 7, 3 + 9, 5 + ) P. (,,, ) cts s the ddtve dentty of P. Now (P, ) s commuttve semgroup of fnte order; (1, 1, 1, 1) s the multplctve dentty of (P, ). Suppose x = (3, +, 8, 5+7 ) nd y = (3+, 8, 9+, 8 ) re n P to fnd x.y = (3, +, 7, 5+7 ) (3 +, 8, 9 +, 8 ) = (3 3+, + 8, 8 (9+ ), ) (mod 1) = ( , , 7 + 8, ) (mod 1) (usng = 11) 1

104 = (9 + 9, +8, 8, + ) s n P. Clerly (P, ) hs zero dvsors, unts, dempotents, dels nd subsemgroups. Interested reder cn study the ssocted propertes of the group of row mtrces wth complex modulo number entres nd semgroup of row mtrces of complex modulo nteger entres. Now let x1 x Y = xm where x s re n C (Z n ); 1 m, tht s x = + b wth = n 1. Y s clled / defned s the column mtrx or column vector of complex modulo ntegers. We see f we get collecton of m 1 column vectors, tht collecton s group under ddton. Infct multplcton s not defned on tht collecton. We wll llustrte ths stuton by some exmples. Exmple.: Let P = x1 x x3 x x5 x C (Z 5 ) = { + b, b Z 5 ; =, 1 5}. P s group under ddton of fnte order. or f x = + 1 nd y =

105 re n P to fnd x + y. x + y = = (mod 5) = s n P. Now cts s the ddtve dentty. Multplcton or product cnnot be defned on P. Exmple.1: Let M = x1 x x x C(Z ) = { + b, b Z }; = 3; 1 } be group of 1 complex modulo nteger column mtrx. M s of fnte order nd s commuttve. Now we proceed to defne m n (m n) complex modulo nteger mtrx nd work wth them. 1

106 Exmple.: Let P = x x x x x x x x x x x x x x x x C (Z ) = { + b, b Z }; = 19; 1 15} be 5 3 complex modulo nteger mtrx. P s group under ddton nd n P we cnnot defne product P s defned s 5 3 mtrx group of complex modulo ntegers. Exmple.3: Let W = C (Z 19 ) = { + b, b Z 19 }; = 18; 1 } be 11 complex modulo nteger group of fnte order under ddton W cts s the ddtve dentty. Exmple.: Let S = C (Z 3 ) = { + b, b Z 3 }; = ; 1 } be group of 5 complex modulo nteger mtrces under ddton. 15

107 Consder P = C(Z 3);1 1 S s subgroup of S of fnte order. Also T = C(Z 3);1 1 S s subgroup of S of fnte order. Now 1 3 P T = C(Z 3);1 5 5 s subgroup of S. Thus lmost ll results true n cse of generl groups cn be esly derved n cse of group of complex modulo nteger mtrx groups. Exmple.5: Let P = 1 3 C (Z 5 ) = { + b, b Z 5 }; = 9; 1 } be the group of complex modulo ntegers under ddton. 1 S = C(Z 5 ); 1 s subgroup of P of fnte order. P 16

108 Exmple.6: Let S = {ll mtrces wth entres from C (Z )} be the group of complex modulo ntegers under ddton whch s commuttve nd s of fnte order. Exmple.7: Let 1 3 M = A 5 = C (Z 7 ) = { + b, b Z 7 }; = 6; 1 9} be the set of 3 3 mtrces. If A, M s group under mtrx multplcton. M s fnte non commuttve group. If we relx the condton A then M s only fnte non commuttve semgroup of complex modulo ntegers. M hs subgroups lso. Also propertes of fnte groups cn be esly extended to the cse of complex modulo ntegers fnte groups wthout ny dffculty. Infct M = {collecton of ll m m squre mtrces wth entres from C (Z n ) (n < )} s non commuttve complex modulo nteger fnte rng of chrcterstc n. Thus we hve nce rng structure on M. So for fnte non commuttve rngs one cn esly mke use of them. We cn hve subrngs, dels, zero dvsors, unts, dempotents nd so on s n cse of usul rngs. Ths work s lso mtter of routne nd hence left s n exercse to the reder. Now hvng seen exmples of rngs, semgroups nd rngs usng mtrces of complex modulo ntegers we proceed onto defne the noton of complex modulo ntegers vector spces nd lner lgebr. DEINITION.3: Let V be complex modulo nteger group under ddton wth entres from C(Z p ); p prme. If V s vector spce over Z p then we defne V to be complex modulo nteger vector spce over Z p. We wll frst llustrte ths stuton by some exmples. 17

109 Exmple.8: Let V = x C (Z 5 ) = { + b, b = Z 5 }; = ; } be complex modulo nteger vector spce over the feld Z 5. Exmple.9: Let x1 x x3 x V = C (Z 3 ) = { + b, b Z 3 }; x5 x 6 = ; 6} be complex modulo nteger vector spce over the feld Z 3. Exmple.3: Let V = C (Z 1 ) = { + b, b Z 1 }; = ; 1 } be complex modulo nteger vector spce over the feld = Z 1. We cn defne subspces, bss, drect sum, pseudo drect sum, lner trnsformton nd lner opertors, whch s mtter of routne so left s n exercse to the reder. However we proceed onto gve some exmples. Exmple.31: Let P =

110 C (Z 3 ) = { + b, b Z 3 }; = ; 1 15} be complex modulo nteger vector spce over the feld Z 3 =. Consder 1 3 M = 5 6 C(Z 3 );1 9 P; M s complex modulo nteger vector subspce of P over = Z S = 3 8 C(Z 3);1 1 P, s gn complex modulo nteger vector subspce of P over Z 3 =. We see 1 M S = 3 C(Z 3);1 6 P 5 6 s gn vector subspce of complex modulo ntegers over the feld Z 3 =. In vew of ths we hve nce theorem the proof of whch s left s n exercse to the reder. THEOREM.11: Let V be complex modulo nteger vector spce over the feld Z p = ; p prme. If W 1, W,, W t (t < ) re complex modulo nteger vector subspces of V over = Z p t then W= W s complex modulo nteger vector subspce of V over. = 1 19

111 Exmple.3: Let V = C (Z 11 ) = { + b, b Z 11 }; = 1; 1 1} be complex modulo nteger vector spce over the feld = Z 11. Consder W 1 = W = 1 C(Z 11);1 3 1 C(Z 11);1 V, V, nd 1 W 3 = C(Z 11);1 3 V 3 1 W = C(Z 11);1 3 V, 3 11

112 be complex modulo nteger vector subspces of V over the feld Z 11. Clerly V = W nd W j W = () f j, 1, j. = 1 Thus V s drect sum or drect unon of complex modulo nteger subspces over Z 11. Consder 1 3 P 1 = C(Z 11);1 3 V, P = 1 3 C(Z 11);1 V, 1 3 P 3 = C(Z 11);1 5 5 V, P = 1 C(Z 11);1 5 3 V, P 5 = 1 3 C(Z 11);1 V 111

113 nd 1 P 6 = C(Z 11);1 V 3 5 be complex modulo nteger vector subspces of V over the complex feld Z 11. Clerly V = 6 P, but P P j () for j; = 1 1, j 6. Thus V s only pseudo drect sum of complex modulo nteger vector subspces of V over Z Exmple.33: Let V = 5 6 C(Z 5 ) = { + b, b Z 5 }; = } be complex modulo nteger vector spce over the feld Z 5. Now consder the set B = 1 1,, 1,,,,, 1 1 1,,,, V. B s bss of V over Z 5. Clerly complex dmenson of V over Z 5 s 1. Lkewse we cn fnd bss for V nd determne the dmenson of V over. Exmple.3: Let 1 3 V = C(Z 13 ) = { + b, b Z 11 }; 11

114 = 1} be complex modulo nteger vector spce over the feld Z 13 nd 1 3 W = 5 6 C(Z 13 ) = { + b, b Z 13 }; = 1} be complex modulo nteger vector spce over the sme feld Z 13. Only now we cn defne the noton of complex lner trnsformton from V to W. Let T : V W be mp such tht T = T s complex lner trnsformton of V to W. It s pertnent to menton here tht most mportnt fctor s f T s complex lner trnsformton we mp to. Of course we cn hve other wys of mppng but those mps should gve us the expected complex lner trnsformton. We cn defne kert = T = V; ker T s gn subspce of V. We lso cn esly derve ll results bout complex lner trnsformtons s n cse of usul trnsformton only wth some smple pproprte opertons on them. We cn lso defne s n cse of usul vector spces the noton of nvrnt subspce by complex lner opertor nd so on. 113

115 Exmple.35: Let V = C (Z 9 ) = { + b, b Z 9 }; = 8; 1 15} be complex modulo nteger vector spce over the feld Z 9 =. Let T : V V be mp such tht T = 3 ; T s lner opertor on V. Now let 1 W = 3 C(Z 9 );1 5 V; 5 W s complex modulo nteger vector subspce of V over Z 9. Now we see T (W) W so W s nvrnt under the complex lner opertor T. Consder 1 3 P = 5 6 C(Z 9 );1 9 V,

116 P s complex modulo nteger vector subspce of V over Z 9 but T (P) P, so P s not complex vector subspce of V whch s nvrnt under the lner opertor T of V. Interested reder cn study the noton of Hom (V, V) nd Hom (W,V) ; p prme. Such study s lso nterestng. Exmple.36: Let V = z p 1 3 z p C(Z 17 ) = { + b, b Z 179 }; = 16} be complex modulo nteger vector spce over the feld Z 17. If 1 W 1 = C(Z 17 ) V, W = C(Z 17 ) V, nd W 3 = C(Z 17 ) 3 V, W = C(Z 17 ) V be subspces of V. We sy V s drect sum of W 1, W, W 3 nd W nd V s spnned by W 1, W, W 3 nd W. We cn hve severl propertes bout lner opertors lke projectons nd the relted results n cse of complex modulo ntegers. We cn defne modulo nteger lner lgebrs f the modulo nteger vector spce s endowed wth product. Infct ll complex modulo nteger lner lgebrs re vector spces but vector spces n generl re not lner lgebrs. Severl of the 115

117 exmples gven prove ths. We now gve exmples of complex modulo nteger lner lgebrs. Exmple.37: Let V = x C (Z 11 ) = { + b, b = Z 11 ; = 1}} be complex modulo nteger lner lgebr over Z 11. Clerly V s of nfnte dmenson over Z 11. Exmple.38: Let M = {(x 1, x, x 3, x, x 5, x 6 ) x C (Z 17 ) = { + b, b Z 17 ; = 16}; 1 6} be complex modulo nteger lner lgebr over Z 17. Clerly M s fnte dmensonl nd M s lso of fnte order. Exmple.39: Let 1 3 V = 5 6 C (Z 53 ) = { + b, b Z 53 }; = 5; 1 9} be complex modulo lner lgebr over Z 53. V s fnte dmensonl nd hs only fnte number of elements n t.however V hs complex modulo nteger lner sublgebrs. Exmple.: Let 1 V = 3 C (Z 61 ) = { + b, b Z 61 ; = 6}; 1 } be complex modulo nteger lner lgebr over the feld = Z 61. Defne T : V V by 1 1 T 3 = ; T s lner opertor on V. urther f 1 W = C(Z 61);1 3 V; 3 116

118 be complex modulo nteger lner sublgebr of V over Z 61 =. T (W) W so T does not keep W s nvrnt subspce of V over Z 61. Exmple.1: Let nd W = V = C(Z 13); C(Z );1 16 be ny two complex modulo nteger lner lgebrs over the feld = Z 13. Defne T : V W by 1 1 T ( 3 ) = ; 3 T s lner trnsformton from V to W. Exmple.: Let V = C (Z 37 ) = { + b, b Z 37 ; = 36}; 1 16} be complex modulo nteger lner lgebr over the feld Z 37 =. 117

119 Defne T : V V by T 3 = ; T s lner opertor on V. All propertes relted wth usul opertors cn be esly derved n cse of complex modulo nteger lner opertors. We cn defne n cse of lner opertors of vector spce / lner lgebr of complex modulo ntegers we cn defne the relted specl chrcterstc vlues nd chrcterstc vectors. Let V be complex modulo nteger vector spce over the feld Z p nd let T be lner opertor on V. A specl chrcterstc vlue of T s sclr c n Z p such tht there s non zero vector α n V wth Tα = cα. If c of chrcterstc vlue of T, then () ny α such tht Tα = cα s clled chrcterstc vector of T ssocted wth vlue c. (b) collecton of ll α such tht Tα = cα s clled the specl chrcterstc spce ssocted wth c. The fct to be remembered s tht n cse of usul vector spces the ssocted mtrx hs ts entres from the feld over whch the spce s defned but n cse of complex modulo ntegers ths my not be possble n generl. We sy n cse of specl chrcterstc equton the roots my not n generl be n the feld Z p but n C (Z p ) where the vector spce s defned hvng ts entres. Thus ths s the mjor dfference between usul vector spce nd the complex modulo nteger vector spce. Now we defne the noton of specl Smrndche complex modulo nteger vector spce / lner lgebr over the S-rng. DEINITION.: Let V be n ddtve beln group wth entres from {C (Z n ) = { + b, b Z n }; = n 1} where Z n s S-rng. If V s vector spce over the S-rng then we defne 118

120 V to be Smrndche complex modulo nteger vector spce over the S-rng Z n. We gve exmples of them. Exmple.3: Let 1 3 V = 5 6 C(Z 1 ) = { + b, b Z 1 ; = 9}; 1 6} be Smrndche complex modulo nteger vector spce over the S-rng R = Z 1. Exmple.: Let V = C(Z 1 ) = { + b, b Z 1 }; = 13} be S-complex modulo nteger vector spce over the rng S = Z 1. Exmple.5: Let V = C(Z 6 ) = { + b, b Z 6 ; = 5}; 1 6} be S-complex modulo nteger lner lgebr over the S-rng S = Z 6. Exmple.6: Let 1 3 M = C(Z 38 ) = { + b, b Z 38 ; = 37}; 1 9} be complex modulo ntegers lner lgebr over Z

121 Tke 1 3 W = 5 6 C (Z 38 ); 1 6} M; W s Smrndche complex modulo nteger lner sublgebr of M over Z 38. Infct M hs mny S-complex modulo nteger lner sublgebrs. Exmple.7: Let V = {(x 1, x, x 3, x, x 5, x 6, x 7, x 8 ) x C(Z 3 )={+b, b Z 3 ; = 33}; 1 8} be Smrndche complex modulo nteger lner lgebr over the S-rng Z 3 = S. Tke P 1 = {(x 1, x,,,,,, ) x 1, x C (Z 3 )} V, P = {(,, x 3,,,,, ) x 3 C (Z 3 )} V, P 3 = {(,,, x, x 5,,, ) x 5, x C (Z 3 )} V, P = {( x 6 ) x 6 C (Z 3 )} V, P 5 = {( x 1 ) x 7 C (Z 3 )} V nd P 6 = {(, x 8 ) x 8 C (Z 3 )} V be smrndche complex modulo nteger lner sublgebrs of V over the S-rng Z 3. Clerly V = 6 P nd P P j = ( = 1 ) f j; 1, j 6. Thus V s drect sum of subspces. Tke M 1 = {(x 1, x,, x 3 ) x C (Z 3 ); 1 3} V, M = {(x 1, x,,, x 3, x ) x C (Z 3 ); 1 } V, M 3 = {(x 1, x x 3 ) x C (Z 3 ); 1 3} V, M = {(x 1, x,, x 3 x ) x C (Z 3 ); 1 } V, M 5 = {(x 1, x 3 x ) x C (Z 3 ); 1 3} V nd M 6 = {(x 1,, x,, x 3 x ) x C (Z 3 ); 1 } V, be S-complex modulo nteger lner sublgebrs of V over the S-rng S = Z 3. 6 V = M ; M M j () f j; 1, j 6. = 1 Thus V s only pseudo drect unon of S-complex modulo nteger sublner lgebrs of V over Z 3 = S. 1

122 Exmple.8: Let 1 V = 3 C(Z 6 ) = { + b, b Z 6 ; = 5}; 1 } be Smrndche complex modulo nteger lner lgebr over the S-rng Z 6. Let 1 W = C(Z 6 );1 3 V, 3 W s Smrndche complex modulo nteger lner sublgebr of V over the S-rng Z 6. Now let 1 P = C(Z 6);1 V, P s complex modulo nteger lner lgebr over the feld = {, 3} Z 6. Thus P s defned s Smrndche pseudo complex modulo nteger lner lgebr over the feld of V. Infct P s lso Smrndche pseudo complex modulo nteger lner sublgebr of V over the feld {,, } Z 6. Also T = C(Z 6 ) V s pseudo Smrndche complex modulo nteger lner sublgebr of V over the feld = {,, } Z 6. Exmple.9: Let V = C (Z 7 ) = { + b, b Z 7 ; = 6}} be complex modulo nteger vector spce over the feld Z 7. Clerly V hs no subvector spce wth entres of the form = x + y; x nd y. Invew of ths we hve the followng theorem. 11

123 THEOREM.1: Let V = C (Z p );... p prme} be the set of ll n n mtrces complex modulo nteger lner lgebr over Z p. ) V hs no complex modulo nteger lner sublgebr wth entres of the mtrx x = + b, nd b, b Z p. x x x x x x ) M = x Z p V s pseudo complex x x x modulo nteger lner sublgebr of V over Z p. The proof s strght forwrd nd hence left s n exercse to the reder. Exmple.5: Let V = C(Z 5) be complex modulo nteger lner lgebr over the feld Z 5. Consder 3 3 X = nd Y = 3 3 n V. Now 3 3 XY = = = 1 1 (usng = nd modulo ddng 5). =

124 Thus f P = Z5 = { Z5} V then P s not lner sublgebr nd product s defned on V but t s not n P. or tke 3 3 x = nd y = 3 3 n P. Now 3 3 x.y = 3 3 = (mod 5) ( 5 = ) = P. Thus P cnnot be lner sublgebr but P cn be complex modulo nteger vector subspce so P s pseudo complex modulo nteger vector subspce of V over Z 5. Exmple.51: Let P = C(Z 1 ) = { 1 + b 1 1, b 1 Z 1 ; = 11}} be S-complex modulo nteger lner lgebr over the S-rng Z 1. P hs pseudo S- complex modulo nteger lner sublgebr gven by S = Z1 P 13

125 s not sublgebr. Thus f C(Z n ) = { + b, b Z n ; = n 1} then Z n C(Z n ) nd Z n s subrng clled the pseudo complex modulo nteger subrng. However Z n ={ Z n, = n 1} s not rng or semgroup under product but only group under ddton for. b (, b Z n ) s b = b (n 1) nd b (n 1) Z n. Thus product s not defned on Z n. Now ll propertes ssocted wth usul vector spces cn be derved n cse of complex modulo nteger vector spces. We just ndcte how lner functonls on complex modulo nteger of lner lgebrs / vector spces defned over Z p s descrbed n the followng. Let 1 3 V = C (Z 3 ) 9 1 = { + b, b Z 3 ; = }; 1 1} be complex modulo nteger vector spce over the feld Z 3. We defne f : V Z 3 s follows: 1 + b1 + b 3 b3 b f b9 1 + b1 = ( 1 + b b b 1 ) mod 3. Thus f s lner functonl on V over Z 3. However we hve to be creful to mke pproprte chnges whle defnng dul spces nd bss propertes ssocted wth lner functonls. urther we cn hve f(v) = even f v ; nd v V. Interested reder cn develop t s mtter of routne. If A = ( + b ) s n n complex modulo nteger mtrx wth entres from Z p, p prme we cn fnd chrcterstc vlues s n cse of usul mtrces. 1

126 Chpter Three NEUTROSOPHIC COMPLEX MODULO INTEGERS In ths chpter for the frst tme the uthors ntroduce the noton of neutrosophc complex modulo ntegers. We work wth these newly defned neutrosophc complex modulo ntegers to bult lgebrc structures. Let C( Z n I ) = { + b + ci + di, b, c, d Z n, = n 1, I = I, (I) = (n 1)I} be the collecton of neutrosophc complex modulo ntegers. Thus neutrosophc complex modulo nteger s -tuple (, b, c, d) where frst coordnte s rel vlue second coordnte sgnfes the complex number, the thrd coordnte the neutrosophc nteger nd the forth coordnte represents the complex neutrosophc nteger. Exmple 3.1: Let C( Z 5 I ) = {( + b + ci + di), b, c, d Z 5, =, I = I, ( I) = I } be the neutrosophc complex modulo nteger. The order of C( Z 5 I ) s 5. Exmple 3.: Let C( Z I ) = {( + b + ci + di), b, c, d Z, = 3, I = I, ( I) = 3I} be the neutrosophc complex modulo ntegers. 15

127 Exmple 3.3: Let C( Z 1 I ) = {( + b + ci + d I), b, c, d Z 1, = 11, I = I, ( I) = 11I} be the neutrosophc complex modulo ntegers. We gve now lgebrc structure usng C( Z n I ). DEINITION 3.1: Let C( Z n I ) = { + b + ci + di, b, c, d Z n, = n 1,I = I, ( I) = (n 1)I}, suppose defne ddton + on C( Z n I ) s follows f x= { + b + ci + di} nd y = {m + p + si + ri} re n C( Z n I ) then x+ y = ( + b + ci + di) + (m + p + si + ri) = (( + m) (mod n) + (b + p) (mod n) + (c + s)i (mod n) + I (d + r) (mod n)) (C( Z n I ), +) s group for = cts s the ddtve dentty. or f x = { + b + ci + d I C( Z n I ) we hve x = (n ) + (n b) + (n c)i + (n d) I C( Z n I ) such tht x + ( x) =. (C( Z n I ), +) s defned s the group of neutrosophc complex modulo ntegers or neutrosophc complex modulo ntegers group. All these groups re of fnte order nd re commuttve. We wll llustrte ths stuton by some exmples. Exmple 3.: Let G = C( Z 11 I ) = {( + b + ci + di), b, c, d Z 11, I = I, ( ) = 1, ( I) = 1I} be neutrosophc complex modulo nteger group under ddton. o(g) = 11. G cn hve only subgroups of order 11 or 11 = 11 or 11 3 only. Exmple 3.5: Let G = {C( Z I ), +} = {, 1,, I, I, 1+, 1 + I, I +, 1 + I, + + I, + I, 1+ + I, I, 1 + I + I, I + + I, 1+ +I + I} be the group neutrosophc complex modulo ntegers of order = 16. We hve H 1 = {, 1}, H = {, I} nd H 3 = {, 1, I, 1 + I} re subgroups of order two nd four respectvely. 16

128 Exmple 3.6: Let 1 3 P = 5 C ( Z 3 I ) 6 = { + b + ci + di, b, c, d Z 3, =, I = I, ( I) = I, 1 6} be group of neutrosophc complex modulo number 3 mtrx under ddton. Exmple 3.7: Let M = 7 x C ( Z I ) = = { + b + ci + di, b, c, d Z, = 1, I = I, ( I) = 1I}; 7} be group of polynomls under ddton of complex neutrosophc modulo ntegers. Clerly M s of fnte order. Exmple 3.8: Let x1 x x3 x P = C ( Z 5 I ) x3 x = { + b + ci + di, b, c, d Z 5, = 9, I = I, ( I) = 9I}} be the group mtrx under ddton wth entres from C( Z 5 I ). Exmple 3.9: Let M = {(x 1, x,, x 11 ) x C( Z 15 I ) = { + b + ci + di, b, c, d Z 15, = 1, I = I, ( I) = 1I}; 1 11} be group of row mtrx of neutrosophc complex modulo ntegers under ddton. Exmple 3.1: Let M = {(,,,,, ) C( Z 11 I ) = { + b + ci + d I, b, c, d Z 11, = 1, I = I, ( I) = 1I}} be row mtrx of neutrosophc complex modulo ntegers. M s group under product. (M does not contn (,,,,, )). 17

129 Exmple 3.11: Let M = C ( Z 13 I ) = { + b+ ci + d I, b, c, d Z 13, = 1, I = I, ( I) = 1I}} be group of neutrosophc complex modulr ntegers under ddton of fnte order. Exmple 3.1: Let b M = c d, b, c, d C ( Z 15 I ) = {m + n + ri + si m, n, r, s Z 15, = 1, I = I, ( I) = 1I}} be group of complex neutrosophc modulo ntegers under ddton. Let b P =,b C( Z15 I ) M; P s subgroup under ddton of M. Tke b W =,b,c C( Z15 I ) c M s lso group under ddton. Consder b B = c d, b, c, d {m + n + ri + si m, n, r, s {, 5, 1} Z 15 }} M s subgroup of M. 18

130 Now we hve seen the concept of subgroups of group. Exmple 3.13: Let M = {( 1,, 1 ) where C( Z I ) = { + b+ ci + di, b, c, d Z, = 39, I = I, ( I) = 39I}; 1 1} be closed under product but s not group under product. But M s commuttve semgroup of neutrosophc complex modulo ntegers under product. M hs dels nd subsemgroups. Exmple 3.1: Let S = C( Z 3 I ) = { + b + ci + di, b, c, d Z 3, =, I = I, ( I) = I}; 1 3} be complex neutrosophc modulo nteger semgroup under ddton. ) nd order of S. b) nd dels f ny n S. c) Cn S hve zero dvsors? d) nd subsemgroups whch re not dels n S. Study () to (d) to understnd bout neutrosophc complex modulo nteger semgroup. Exmple 3.15: Let b M = c d, b, c, d C ( Z 13 I ) = {m +r + ni + Is = 1, I = I, nd (I ) = 1I} be neutrosophc complex modulo nteger semgroup under product of mtrces.. nd order of M.. nd subsemgroups whch re not dels of M.. Is P =,b C( Z13 I ) b M n del of M? 19

131 v. nd rght dels n M, whch re not left dels of M. v. Prove M s non-commuttve semgroup. v. Is M S-semgroup? Answer such questons to understnd ths concept. However the lst chpter of ths book provdes mny problems for the reder. urther we re not nterested n the study of complex neutrosophc modulo nteger semgroups or groups. Wht we re nterested s the constructon nd study of complex neutrosophc modulo nteger vector spces / lner lgebr nd set complex neutrosophc modulo nteger vector spce / lner lgebr nd ther prtculrzed forms. Now we hve seen exmples of semgroups nd groups bult usng the complex neutrosophc modulo ntegers. We wll frst defne vector spces of complex neutrosophc modulo ntegers. DEINITION 3.: Let V be n ddtve beln group of complex neutrosophc modulo ntegers usng C( Z p I ), p prme. Z p be the feld. If V s vector spce over Z p then V s defned s the complex neutrosophc modulo nteger vector spce over Z p. We wll llustrte ths stuton by n exmple. Exmple 3.16: Let V = 5 x C( Z 7 I ) = = { + b + ci + di, b, c, d Z 7 }, = 6, I = I}; 5} be complex neutrosophc modulo nteger vector spce over the feld Z 7. Exmple 3.17: Let 1 3 M = C ( Z 11 I ) 13

132 = { + b + ci + di, b, c, d Z 11 }, = 1, I = I; ( I) = 1I}; 1 9} be neutrosophc complex modulo nteger vector spce over the feld Z 11. Exmple 3.18: Let 1 P = C( Z 3 I ) 1 = { + b + ci + di, b, c, d Z 3 }, =, I = I; ( I) = I}; 1 1} be complex neutrosophc modulo nteger vector spce over the feld = Z 3. Exmple 3.19: Let M = {( 1,,, 1 ) C ( Z 7 I ) = { + b + ci + di, b, c, d Z 7 }, = 6, I = I; ( I) = 6I}; 1 1} be neutrosophc complex modulo nteger vector spce over the feld = Z 7. Exmple 3.: Let W = C( Z 3 I ) = { + b + ci + di, b, c, d Z 3 }, =, I = I; ( I) = I}; 1 } be neutrosophc complex modulo nteger vector spce over the feld Z 3 =. Exmple 3.1: Let p1 p p3 p p5 p 6 W = p C( Z 5 I ) p3 p p5 = { + b + ci + di, b, c, d Z 5 }, =, I = I; ( I) = I}; 1 5} be neutrosophc complex modulo nteger vector spce over the feld = Z

133 Exmple 3.: Let V = C ( Z 9 I ) = { + b + ci + di, b, c, d Z 9 }, = 8, I = I; ( I) = 8I}; 1 1} be the neutrosophc complex modulo nteger vector spce over the feld Z 9 =. The defnton of subvector spce s mtter of routne we gve few exmples of them. Exmple 3.3: Let V = C ( Z 13 I ) = { + b + ci + di, b, c, d Z 13 }, = 1, I = I; ( I) = 1I}; 1 } be the neutrosophc complex modulo nteger vector spce over Z 13. Consder T = < > 11 1 C( Z 13 I );1 11 V, T s complex neutrosophc modulo nteger vector subspce of V over Z

134 133 M = C( Z I );1 1 < > V be neutrosophc complex modulo nteger vector subspce of V over Z 13. Now T M = C( Z I );1 6 < > V s subspce of V. We hve the followng theorem the proof of whch s left s n exercse to the reder. THEOREM 3.1: Let V be neutrosophc complex modulo nteger vector spce over the feld = Z p. Let W 1,, W t (t < ) be the collecton of vector subspces of V.

135 W = W s vector subspce of V. W cn be the zero subspce. Exmple 3.: Let 1 3 V = C ( Z 7 I ) = { + b + ci + di, b, c, d Z 7 ; = 6, I = I; ( I) = 6I}; 1 1} be neutrosophc complex modulo nteger vector spce over Z 7. Consder W 1 = 1 3 C( < Z7 I > );1 3 V, W = 1 C( < Z7 I > );1 3 3 V, W 3 = W = 1 1, C( < Z7 I > ) 1, C( < Z7 I > ) 1 V, V nd W 5 = 1 1, C( < Z7 I > ) V 13

136 be neutrosophc complex modulo nteger vector subspces of V over Z 7. Clerly V = drect sum of W 1, W,, W 5. 5 W f j, 1, j 5. Thus V s the = 1 Exmple 3.5: Let V = C( Z 53 I ) = { + b + ci + di, b, c, d Z 53 ; = 5, I = I; ( I) = 5I}; 1 8} be complex neutrosophc modulo nteger vector spce over the feld = Z 53. Exmple 3.6: Let M = C(Z 53 ) = { + b = 5} 1 8} V, (V mentoned n Exmple 3.5) M s clled s the complex pseudo neutrosophc complex modulo nteger vector subspce of V over Z 53. Tke P = Z53 I = { + bi I = I,, b Z 53 } V. P s neutrosophc pseudo neutrosophc complex modulo nteger vector subspce of V over Z

137 Now f T = Z 53;1 8 V; then T s rel pseudo neutrosophc modulo nteger vector subspce of V over Z 53. It s nterestng to note tht V s not lner lgebr. We cn lso hve B = Z I {I Z }; = 53 V s pure neutrosophc pseudo neutrosophc modulo nteger vector subspce of V over Z 53. nlly f we tke C = j Z53 ;{ Z53}; 1 j V s gn pure complex pseudo neutrosophc complex modulo nteger vector subspce. However t s pertnent to menton here tht f V s lner lgebr then V cnnot contn pure complex pseudo neutrosophc complex modulo nteger vector subspce. Ths s the one of the mrked dfference between vector spce nd lner lgebr. We hve not so fr defned the noton of neutrosophc complex modulo nteger lner lgebr. We just sy complex neutrosophc modulo nteger vector spce V s complex neutrosophc modulo nteger lner lgebr of V s closed under product. Invew of ths we hve the followng theorem. 136

138 THEOREM 3.: Let V be complex neutrosophc modulo nteger lner lgebr over the feld Z p, then V s neutrosophc complex modulo nteger vector spce over the feld Z p. If V s complex neutrosophc modulo nteger vector spce then V need not n generl be lner lgebr. The frst prt follows from the very defnton nd the lter prt s proved by counter exmple. Exmple 3.7: Let M = C ( Z 17 I ) = { + b + ci + di, b, c, d Z 17 ; = 16, I = I; ( I) = 16I}; 1 1} be complex neutrosophc modulo vector spce over the feld Z 17. Clerly product cnnot be defned on M so M s only vector spce nd not lner lgebr. Exmple 3.8: Let M = C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3 ; =, I = I; ( I) = I}; 1 16} be complex neutrosophc modulo nteger lner lgebr over the feld Z 3. We cn defne sublgebrs nd drect sum. Exmple 3.9: Let M = ci + di, b, c, d Z 11 }, x C ( Z 11 I ) = { + b + = =, I = I nd (I ) = 1I} be 137

139 complex neutrosophc modulo nteger lner lgebr of nfnte dmenson over Z 11. Exmple 3.3: Let V = {( 1,, 3,, 5, 6, 7, 8 ) C ( Z 17 I ) = { + b + ci + di, b, c, d Z 17 }, = 16, I = I nd ( I) = 16I}; 1 8} be neutrosophc complex modulo nteger lner lgebr over the feld = Z 17. Clerly V s fnte dmensonl over Z 17. Consder P 1 = {(x 1,, x,,,,, ) x 1, x C( Z 17 I )} V, P = {(, x 1,, x,,,, ) x 1, x C( Z 17 I )} V, P 3 = {(,,,, x 1, x,, ) x 1, x C( Z 17 I )} V nd P = {( x 1 x ) x 1, x C ( Z 1 I )} V be collecton of complex neutrosophc modulo nteger lner sublgebrs of V over Z 17. Clerly V = P ; P P j = ( = 1 ) f j, 1, j. Thus V s drect sum of lner sublgebrs. Tke N = {(x 1, x, x 3, x, x 5, x 6, x 7, x 8 ) x C(Z 17 ) nd x = ; Z 17 ; = 16} V; clerly N s pseudo complex neutrosophc complex modulo nteger pseudo vector subspce of V over Z 17. Clerly N s not pseudo complex neutrosophc complex modulo lner sublgebr of V over Z 17. Tke B = {(x 1, x, x 3, x, x 5, x 6, x 7, x 8 ) x C(Z 17 ) where x = + b wth, b Z 17 ; = 16} B; clerly B s pseudo complex neutrosophc complex modulo nteger lner sublgebr of V over Z 17. or f x = (,,, 8, 3,,, 8 ) s n N we see x.y = (,,, 8, 3,,, 8 ). (3, 9,,,, 8,, 9 ) = (6.16,,, 8, 3,, 8,., 8, 9 ) = (11,, 16,,, 16, ) N. Thus N s not closed under product so N s not pure complex modulo nteger lner lgebr over Z 17. Let A = {(x 1, x,, x 8 ) x Z 17 I = { + bi, b Z 17, I = I; 1 8} V; V s pseudo neutrosophc complex neutrosophc modulo nteger sublner lgebr of V over Z

140 Consder B = {(x 1, x,, x 8 ) x Z 17 I = {I Z 17, I = I; 1 8} V be pure neutrosophc pseudo neutrosophc complex modulo nteger lner sublgebr of V over Z 17. Now we cn proceed onto defne other propertes whch cn be treted s mtter of routne we would provde more llustrte exmples. Exmple 3.31: Let M = C ( Z 11 I ) = { + b + ci + di, b, c, d Z 11 ; = 1, I = I; ( I) = 1I}; 1 1} nd N = C ( Z 11 I ) = { + b + ci + di, b, c, d Z 11 ; = 1, I = I; ( I) = 1I}; 1 1} be two neutrosophc complex modulo nteger vector spces over Z 11. Consder T: M N defned by T =

141 for every M T s lner trnsformton of M to N. We cn defne severl other lner trnsformtons from M to N. Exmple 3.3: Let M = C ( Z 19 I ) = { + b + ci + di, b, c, d Z 19 ; = 18, I = I; ( I) = 18I}; 1 1} be neutrosophc complex modulo nteger vector spce over the feld Z 19. Defne T : M M by T 5 = ; T s lner opertor on M. Severl such lner opertors cn be defned. Suppose 1 3 W = ( < Z19 I > );1 6 M 5 6 be complex neutrosophc modulo nteger vector subspce of M over Z 19. We see T (W) W; thus W s nvrnt subspce of M over Z 19. 1

142 Consder 1 3 P = C( < Z19 I > );1 6 M; 5 6 P s complex neutrosophc modulo nteger vector subspce of M over Z 19. We see T = Thus P s lso nvrnt under T. Consder η : M M gven by η = 6, η s lner opertor on M. But η (W) W so W s not nvrnt under η. urther η (P) P so P s lso not nvrnt under η, we cn derve severl propertes lke reltng nullty nd rnk T wth dmenson of V, V complex neutrosophc modulo nteger vector spce nd T lner trnsformton from V to W; W lso complex neutrosophc modulo nteger vector spce over the sme feld s tht of V. We cn combne two lner opertors nd fnd Hom (V, V) nd so on. Zp Exmple 3.33: Let V = C ( Z 13 I ); 1 } be complex neutrosophc modulo nteger vector spce over the feld Z

143 Defne f : V Z 13 by f f s lner functonl on V. Re prt of (mod13) ; = = 1 Exmple 3.3: Let V = {( 1, ) C ( Z 7 I ) = { + b + ci + di, b, c, d Z 7 ; = 6, I = I; ( I) = 6I}; 1 } be neutrosophc complex modulo nteger vector spce over the feld Z 7. We defne f : V Z 7 by f(( 1, )) = Reprt of (mod7). = 1 or nstnce f (( I + I), (6+ +.I + 3 I)) = (mod 7) = (mod 7). Thus f s lner functonl on V. Suppose nsted of defnng neutrosophc complex modulo nteger vector spce V over feld Z p f we defne V over Z n, Z n S-rng then we cll V s Smrndche neutrosophc complex modulo nteger vector spce over the S-rng Z n. We wll gve exmples of ths stuton. Exmple 3.35: Let V = C ( Z 78 I ) = { + b + ci + di, b, c, d Z 78 }, = 77, I = I nd ( I) = 77I}, 1 } be Smrndche complex neutrosophc modulo nteger vector spce over the S-rng Z 78. Exmple 3.36: Let V = C ( Z 9 I ) 1

144 = { + b + ci + di, b, c, d Z 9 }, = 93, I = I nd ( I) = 93I}, 1 3} be the S-neutrosophc complex modulo nteger vector spce over the S-rng Z 9. Exmple 3.37: Let P = {( 1,, 3,,, ) C( Z 1 I ) = { + b + ci + di, b, c, d Z 1 }, = 9, I = I nd ( I) = 9I}, 1 } be S-neutrosophc complex modulo nteger lner lgebr over the S-rng Z 1. Exmple 3.38: Let M = C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3 }, = 33, I = I nd ( I) = 33I}, 1 1} be S-neutrosophc complex modulo nteger vector spce over the S-rng Z 3. It s mportnt nd nterestng to note tht every S- neutrosophc complex modulo nteger vector spce over the S-rng s n generl not S-neutrosophc complex modulo nteger lner lgebr but lwys S-neutrosophc complex modulo nteger lner lgebr over S-rng s vector spce over S-rng. Exmple 3.39: Let V = C ( Z 6 I ) = { + b + ci + di, b, c, d Z 6 }, = 5, I = I nd ( I) = 5I}, 1 18} be S-neutrosophc complex modulo nteger vector spce over the S-rng Z 6. Clerly V s not S-neutrosophc complex modulo nteger lner lgebr over Z 6. 13

145 Exmple 3.: Let V = C ( Z 1 I ) = { + b + ci + di, b, c, d Z 1 }, = 13, I = I nd ( I) = 13I}, 1 7} be Smrndche complex neutrosophc modulo nteger vector spce over the S-rng Z 1. Consder 1... P =... 1, ( Z1 I ) V... s Smrndche complex neutrosophc modulo nteger vector subspce of V over Z 1. Tke... 1 M =... 5 C( Z1 I );1 6 V s Smrndche complex neutrosophc modulo nteger vector subspce of V over Z P M =... V... s the zero subspce of V over Z 1. Exmple 3.1: Let P = C ( Z 6 I ) 1

146 = { + b + ci + di, b, c, d Z 6 }, = 5, I = I nd ( I) = 5I}, 1 18} be S-neutrosophc complex modulo nteger vector spce of over Z 6. Consder 1 3 W 1 = 1,, 3 C ( Z I ) = { + b + ci + di, b, c, d Z }, = 1, I = I nd ( I) = 1I}} V, W = W 3 = ,, 3 C ( Z I )} V, 1,, 3 C ( Z I )} V, W = 1 3 1,, 3 C ( Z I )} V, 15

147 nd W 5 = 1 3 W 6 = 1 3 1,, 3 C ( Z I )} V 1,, 3 C ( Z I )} V be S-neutrosophc complex modulo number vector subspces of V. V = 6 W ; W W j = () f j; 1, j 6. Thus V s the = 1 drect sum of S-neutrosophc complex modulo nteger vector subspces of V over Z. Exmple 3.: Let 1 3 V = 5 6 C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3 }, = 33, I = I nd ( I) = 33I}, 1 9} be Smrndche neutrosophc complex modulo nteger vector spce over the S-rng Z 3. Tke 1 W 1 = 3 1,, 3 C ( Z 3 I )} V, 16

148 W = W 3 = ,, 3 C ( Z 3 I )} V, 1,, 3 C ( Z 3 I )} V, nd W = W 5 = ,, 3 C ( Z 3 I )} V, 1,, 3 C ( Z 3 I )} V be S-neutrosophc complex modulo nteger vector subspces of V over the S-rng Z 3. Clerly 5 W ; W W j = 1 f j; 1 j, j 5. So V s not drect sum of W 1, W,, W 5, but only pseudo drect sum of W 1, W,, W 5 of V. We cn lso defne neutrosophc complex modulo nteger vector spce / lner lgebr over Z n I. We wll gve only exmples of them. Exmple 3.3: Let V = C ( Z 9 I ) 17

149 = { + b + ci + di, b, c, d Z 9 }, = 8, I = I nd ( I) = 8I}, 1 16} be neutrosophc complex neutrosophc modulo nteger lner lgebr over = C ( Z 9 I ). Exmple 3.: Let V = C ( Z I ) = { + b + ci + di, b, c, d Z }, = 39, I = I nd ( I) = 39I}, 1 33} be neutrosophc complex neutrosophc modulo nteger vector spce over ( Z I ). Exmple 3.5: Let V = C( Z 9 I ) = { + b + ci + di, b, c, d Z 9 }, = 8, I = I nd ( I) = 8I}, 1 } be complex neutrosophc complex modulo nteger vector spce over the complex rng C(Z 9 ) = { + b, b Z 9 }, = 8}. Exmple 3.6: Let V = C( Z 1 I ) = { + b + ci + di, b, c, d Z 1 }, = 11, I = I nd ( I) = 11I}, 1 16} be complex neutrosophc complex modulo nteger lner lgebr over the complex rng C (Z 1 ) = { + b, b Z 1 }, = 11}. 18

150 Exmple 3.7: Let V = C ( Z 16 I ) = { + b + ci + di, b, c, d Z 16 }, = 15, I = I nd ( I) = 15I}, 1 } be complex neutrosophc complex modulo nteger vector spce over the complex rng C (Z 16 ) = { + b, b Z 16 }, = 15}. Exmple 3.8: Let 1 3 V = C ( Z 1 I ) = { + b + ci + di, b, c, d Z 1 }, = 13, I = I nd ( I) = 13I}, 1 1} be strong neutrosophc complex modulo nteger vector spce over the complex neutrosophc modulo nteger rng C ( Z 1 I ). Clerly dmenson of V over S s 1. Exmple 3.9: Let V = C ( Z 13 I ) = { + b + ci + di, b, c, d Z 13 }, = 1, I = I nd ( I) = 1I}, 1 } be strong neutrosophc complex modulo 19

151 nteger vector spce over the modulo nteger rng C ( Z 13 I ). V s fnte dmensonl nd dmenson of V over S s. Exmple 3.5: Let 1 3 V = C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3 }, =, I = I nd ( I) = I}, 1 9} be specl complex neutrosophc modulo nteger vector spce over the neutrosophc complex modulo nteger rng S = C ( Z 3 I ). Consder B = 1 1 1,,, 1, 1, 1,,, V, B s bss of V over S. Clerly dmenson of V over S s nne. However f S s replced by Z 3 I or Z 3 or C (Z 3 ) the dmenson s dfferent from nne. Exmple 3.51: Let V = C ( Z 37 I ) 15

152 = { + b + ci + di, b, c, d Z 37 }, = 36, I = I nd ( I) = 36I}} be strong neutrosophc complex modulo nteger vector spce over the rng S = C ( Z 37 I ). Tke 1 3 W = C ( Z 37 I ); 1 8} V, W s strong neutrosophc complex modulo nteger vector spce. Consder P = 1 3 C (Z 37 ) = { + b, b C(Z 37 )}, = 36, C ( Z 37 I )}; 1 } V, P s only pseudo strong complex neutrosophc complex modulo nteger vector subspce of V over C (Z 37 ). Clerly P s not strong vector subspce of V over C ( Z 37 I ). Consder 1 B = 3 C ( Z 37 I ) 6 5 = { + b, b Z 37 }, I = I,; 1 6} V; B s only pseudo neutrosophc strong neutrosophc complex modulo nteger vector subspce of V over the neutrosophc feld Z 37 I C( Z 37 I ). Clerly B s not strong vector subspce of V over C ( Z 37 I ). Consder 1 3 C = Z 37 ; 1 1}

153 V, C s only usul modulo nteger vector spce over Z 3. Thus C s pseudo strong usul modulo nteger neutrosophc complex vector subspce of V over Z 37 C( Z 37 I ). Clerly C s not strong vector spce over C( Z 37 I ). Thus we hve the followng theorem. THEOREM 3.3: Let V be strong neutrosophc complex modulo nteger vector spce / lner lgebr over C ( Z p I ). Then. V hs pseudo complex strong neutrosophc complex modulo nteger vector subspce / lner sublgebr over C (Z p ) = { + b, b Z p, = p 1 C ( Z p I ).. V hs pseudo neutrosophc strong neutrosophc complex modulo nteger vector subspce / lner sublgebr over Z p I = { + bi, b Z p, I = I} C ( Z p I ).. V hs pseudo rel modulo nteger strong neutrosophc complex vector subspce / lner sublgebr over Z p C ( Z p I ). Proof s smple nd hence s left s n exercse to the reder. Exmple 3.5: Let V = 1 3 C ( Z I ) = { + b + ci + di = 1, I = I nd ( I) = I}, 1 } be strong neutrosophc complex modulo nteger lner lgebr over the neutrosophc complex modulo nteger rng. C( Z I ) M = 1 3 C (Z ); 1 } V be pseudo rel strong neutrosophc complex modulo nteger lner sublgebr of V over Z C( Z I ). Clerly M s not strong neutrosophc complex modulo nteger lner sublgebr of V over C( Z I ). 15

154 Consder P = 1 3 C (Z ); { + b = 1,, b Z }; 1 } V; P s pseudo complex strong neutrosophc complex modulo nteger lner sublgebr of V over C(Z ) C( Z I ). Clerly P s not strong complex neutrosophc modulo nteger lner sublgebr of V over C( Z I ). 1 B = 3 C ( Z I ) = { + bi, b Z, I = I} C( Z I ); 1 } V, s pseudo neutrosophc strong complex neutrosophc complex modulo nteger lner sublgebr of V over Z I. Clerly B s not strong complex neutrosophc modulo nteger lner sublgebr of V over C( Z I ). Now we proceed onto defne the noton of set neutrosophc complex modulo nteger vector spce over set. DEINITION 3.3: Let V be set of elements from C ( Z n I ) (the elements cn be mtrces wth entres from C ( Z n I ) or polynoml wth coeffcents from C ( Z n I ). Suppose S Z n be subset of Z n. If for ll v V nd s S, sv = vs V then we defne V to be set neutrosophc complex modulo nteger vector spce over the set S Z n. We wll gve exmples of ths stuton. Exmple 3.53: Let V = 1 x, (x 1, x, x 3, x, x 5 ), = x1 x 1 3, 5 6 x 7, x j C( Z 7 I ); 1, 1 j 7} be set complex neutrosophc vector spce over the set {, 1, 5} Z

155 Exmple 3.5: Let V = x1 x x3 x1 x... x1 x1 x x 3 x x5 x 6,, x11 x 1... x x x5 x 6 x1 x... x 3 x8 x9 x3 x C( Z 19 I ); 1 3} be set neutrosophc complex modulo nteger vector spce over the set S = {, 1, 11, 1,, 7} Z 19. Exmple 3.55: Let V =,, C( Z 7 I );{ + b + ci + di, b, c, d Z 7 }, = 6, I = I nd ( I) = 6I}, 1 9} be set neutrosophc complex modulo nteger vector spce over the set S = {, 1, 7, 16, 19,, 3} Z 7. Exmple 3.56: Let V = 1 1,,(,,..., ) 1 19 C( Z 19 I ); { + b + ci + di, b, c, d Z 19 }, = 18, I = I nd ( I) = 18 }, 1 8} be set complex neutrosophc modulo nteger vector spce over the set S = {, 1,, 1, 1, 5, 16} Z

156 Let us now gve exmples of set complex neutrosophc vector subspces nd subset neutrosophc complex modulr nteger subspces of vector spce. The defnton s left s n exercse. Exmple 3.57: Let V = x,, = , , C ( Z 19 I );{ + b + ci + di, b, c, d Z 19 }, = 18, I = I nd ( I) = 18I}, 1 } be set neutrosophc complex modulo nteger vector spce over the set S = {I,, 1,, + 3I, 18, 15I, 6I, 8 + 5I} Z 19 I. Consder P = x,, = , 155

157 1 1 3, C( Z 19 I ); 1 1} V, P s set complex neutrosophc modulo nteger vector subspce of V over the set S. Consder = 6 8 B = x,,, 1, 1 3 C ( Z 19 I ); 1 9} V, B s subset complex neutrosophc vector subspce of V over the subset S = {, 1,, 18, 6I, 8 + 5I} Z 19 I. 156

158 Exmple 3.58: Let V =,, x 3 C ( Z 3 I ) = = { + b + ci + di, b, c, d Z 3 }, =, I = I nd ( I) = I}, 3} be set neutrosophc complex modulo nteger vector spce over the set S = {,,, 19, I} C( Z 3 I ). Tke 1 1 P =, x 3, b, c, d, C( Z 3 I ); 1} = V; P s set neutrosophc complex modulo nteger vector subspce of V over S. Exmple 3.59: Let V =, C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3 }, =, I = I nd ( I) = I}, 1 3} be set neutrosophc complex modulo nteger vector spce over the set S = {, 1, 5,, 18, 7} Z 3. Tke M =,,... 3 C( Z 3 I ); 1 6} V; M s set complex neutrosophc modulo nteger vector subspce of V over S. 157

159 Now we provde exmples of set neutrosophc complex modulo nteger lner lgebr over the set S. Exmple 3.6: Let V =,, x 3 6 = C ( Z 11 I ) = { + b + ci + di, b, c, d Z 11 }, = 1, I = I nd ( I) = 1I}, 7} be set neutrosophc complex modulo nteger vector spce over the set S = {1,,, 1+3I, I, I+1}. Consder B = , x, 3 = C ( Z 11 I ); 1} V, B s set neutrosophc complex modulo nteger vector subspce over the set S. Now we see n exmple 3.6 we cnnot defne ddton on V. So V s not lner lgebr. Exmple 3.61: Let V = 5 x C( Z 17 I ) = { + b + = ci + di, b, c, d Z 17 }, = 16, I = I nd ( I) = 16I}, 5} be set neutrosophc complex modulo nteger lner lgebr over the set S = {, 1,, 1, 1}. Exmple 3.6: Let V = 5 x C ( Z 17 I ) = { + b + = ci + di, b, c, d Z 17 }, = 16, I = I nd ( I) = 16I}, 5} be set neutrosophc complex modulo nteger lner lgebr over the set S = {, 1,, 1, 1}. 158

160 Exmple 3.63: Let B = C( Z 37 I ) = { + b + ci + di, b, c, d Z 37 }, = 36, I = I nd ( I) = 36I},1 3} be set neutrosophc complex modulo nteger lner lgebr over the set S = {, 1,, 3, 1, 18, 3, 31}. Exmple 3.6: Let V = {( 1,,, 13 ) C( Z 31 I ) = { + b + ci + di, b, c, d Z 31 }, = 3, I = I nd ( I) = 3I}, 1 31} be set neutrosophc complex modulo nteger lner lgebr over the set S = {, 1, 3}. We gve exmples of lner sublgebrs. 1 3 Exmple 3.65: Let V = C ( Z 3 I ) = { + b + ci + di, b, c, d Z }, =, I = I nd ( I) = I}, 1 9} be set complex neutrosophc modulo nteger lner lgebr over the set S = {,, 8,, 9, 1, 39}. We see on V we cn defne yet nother operton product; so V becomes set neutrosophc complex modulo nteger strong lner lgebr over S. We hve the followng nterestng observtons relted wth such the set lgebrs. THEOREM 3.: Let V be set complex neutrosophc modulo nteger strong lner lgebr over the set S. () V s set complex neutrosophc modulo nteger lner lgebr over the set S. () V s set complex neutrosophc modulo nteger vector spce over the set S. However the converse of both () nd () re not true n generl. 159

161 Exmple 3.66: Let V = C( Z 3 I ) = { + b + ci + di, b, c, d Z 3 }, =, I = I nd ( I) = I}, 1 1} be set complex neutrosophc modulo nteger lner lgebr over the set S = {I,, 1 + I, 5}. Clerly V s not set complex neutrosophc modulo nteger strong lner lgebr over the set S. However V s set complex neutrosophc modulo nteger vector spce over the set S. Ths proves the converse prt for the clm (1) of theorem 3. Exmple 3.67: Let V,( 1,, 3,, 5, 6, 7, 8, 9), = C( Z 11 I )= { + b + ci + di, b, c, d Z 3 }, = 1, I = I nd ( I) = 11I}, 1 3} be set complex neutrosophc modulo nteger vector spce over the set S = {, I, 1+I, 5+6I, 1I + 3}. Clerly V s not set complex neutrosophc modulo nteger lner lgebr over the set S. urther t s not strong lner lgebr s t s not even lner lgebr. Hence converse of () of theorem s verfed. Exmple 3.68: Let V = {( 1,,, 11 ); where C ( Z 67 I )= { + b + ci + di, b, c, d Z 67 }, = 66, I = I nd ( I) = 66I}, 1 11} be set complex neutrosophc modulo nteger strong lner lgebr over the set S = {, 1, I, + 33I, + I, 5+17I, I + 1}. Clerly V s set complex neutrosophc modulo nteger lner lgebr over the set S. V s lso set complex neutrosophc modulo nteger vector spce over S. 16

162 Now we wll gve exmples of these structures. Exmple 3.69: Let 1 V = 3 C ( Z 13 I ) = { + b + ci + di, b, c, d Z 13 }, = 1, I = I nd ( I) = 1I}} be set complex neutrosophc modulo nteger strong lner lgebr over the set S = {, 1,, 5, 7, 1, }. 1 W = C ( Z 13 I ); 1 3} V 3 s set complex neutrosophc modulo nteger strong lner sublgebr of V over S. 1 P = C ( Z 13 I ); 1 } V s lso set complex neutrosophc modulo nteger strong lner sublgebr of V over S. Tke 1 M = C ( Z 13 I ); 1 } V, M s only pseudo set complex neutrosophc strong lner sublgebr of V over S s M s not strong set lner sublgebr of V over S s product s not defned on M. If we tke 1 3 P =,, 1,, 3 C ( Z 13 I )} M, P s only pseudo set complex neutrosophc modulo nteger subspce of V over S s n P we see the sum of two elements. 1 x = nd y 3 s x + y = = P. Hence the clm. 161

163 Exmple 3.7: Let 1 3 V = 5 6 1,, 3 C( Z 19 I ) = { + b + ci + di, b, c, d Z 19 }, = 18, I = I nd ( I) = 18I}} be set complex neutrosophc modulo nteger strong lner lgebr over the set S = {, 1, I+3, 9+8I, 3, I}. Consder P 1 = ( Z19 I ) V nd P = b ( Z19 I ) V b be set complex neutrosophc modulo nteger strong lner sublgebr over S of V. However we my not be n poston to wrte V s drect sum of strong lner sublgebrs. Exmple 3.71: Let V = {( 1,, 3,, 5, 6, 7, 8 ) C ( Z 5 I ) = { + b + ci + di, b, c, d Z 59 }, =, I = I nd ( I) = I}} be set neutrosophc complex modulo nteger strong lner lgebr over the set S = {, 1, I + 3, I + }. Consder P 1 = {( 1,, ) 1 C ( Z 5 I )} V, P = {(,, 1,, ) 1, C ( Z 5 I )} V; P 3 = {(,,,, 1,, ) 1, C ( Z 5 I )} V nd P = {(, 1, ) 1, C ( Z 5 I )} V; P 1, P, P 3 nd P re set complex neutrosophc modulo nteger strong lner sublgebrs of V over the set S. Clerly V = Clerly P ; P P j = ( ) f = 1 j; 1 j, j. V s drect sum of strong lner sublgebrs over the set S. Let us consder the followng exmples. 16

164 Exmple 3.7: Let 1 3 V = 5 6 C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3 }, =, I = I nd ( I) = I}; 1 9} be set complex neutrosophc modulo nteger strong lner lgebr over the set S = {, 1, 1+I, +I, }. Consder 1 P 1 = C( Z3 I );1 3 V, 3 1 P = 3 C( Z3 I );1 3 V nd P 3 = 1 C( Z3 I );1 3 V 3 re only pseudo set complex neutrosophc modulo nteger strong lner sublgebrs of V. They re pseudo strong s they re not closed under product, tht s product s not closed on P 1, P nd P 3. But we see V = 3 P ; P P j = () f j; 1 j, j 3. Thus = 1 V s only pseudo drect sum nd not drect sum of strong lner sublgebrs. Now hvng seen propertes one cn defne lner trnsformtons provded they re defned on the sme set. Exmple 3.73: Let 1 3 V = C ( Z 13 I ) 163

165 = { + b + ci + di, b, c, d Z 3 }, = 1, I = I nd ( I) = 1I}; 1 9} be set neutrosophc complex modulo nteger strong lner lgebr defned over the set S = {, I, I, + 7I, 9I, 8, 1 + 3I}. Consder M = C ( Z 13 I ) = { + b + ci + di, b, c, d Z 13 }, = 1, I = I nd ( I) = 1I};1 16} be set neutrosophc complex modulo nteger strong lner lgebr over the sme set S. We defne T : V M by T = T s strong lner trnsformton from V to M. Suppose we defne P : V M by P = , P s not strong lner trnsformton from V to M only lner trnsformton from V to M. We cn derve ll relted propertes wth pproprte modfctons. Ths tsk s left s n exercse to the reder. Now we proceed onto defne the noton of semgroup neutrosophc complex modulo nteger vector spces / lner lgebrs over semgroup. 16

166 DEINITION 3.: Let V be set neutrosophc complex modulo nteger vector spce over the set S. If S s n ddtve semgroup then we defne V to be semgroup complex neutrosophc modulo nteger vector spce over the semgroup S. We wll gve exmples of ths stuton. Exmple 3.7: Let V = x, , C ( Z 19 I ) = = { + b + ci + di, b, c, d Z 19 }, = 18, I = I nd ( I) = 18I}; 3} be semgroup complex neutrosophc modulo nteger vector spce over the semgroup Z 19. Exmple 3.75: Let V = ,( 1,,..., ), x, = C ( Z 61 I )= { + b + ci + di, b, c, d Z 19 }, = 6, I = I nd ( I) = 6I}; 33} be semgroup complex neutrosophc modulo nteger vector spce over the semgroup C (Z 61 ) = { + b, b Z 61 }, = 6}. Exmple 3.76: Let V, , 5 = 6,( 1,,..., 5 )

167 C ( Z 3 I )= { + b + ci + di =, I = I nd ( I) = I}; 1 } be semgroup neutrosophc complex modulo nteger vector spce over the semgroup S = (Z 3 ). We cn defne semgroup complex neutrosophc modulo nteger lner lgebr over semgroup S under ddton. We gve only exmples of them. Exmple 3.77: Let V = C ( Z 9 I ) ={ + b + ci + di, b, c, d Z 9 }, = 8, I = I nd ( I) = 8I}; 1 } be semgroup neutrosophc complex modulo nteger lner lgebr over the semgroup Z 9 = S. Exmple 3.78: Let V = C ( Z 17 I ) ={ + b + ci + di, b, c, d Z 16 }, = 16, I = I nd ( I) = 16I}; 1 8} be semgroup neutrosophc complex modulo nteger lner lgebr over the semgroup S = Z 17. Exmple 3.79: Let V = C ( Z 7 I ) 166

168 = { + b + ci + di, b, c, d Z 7 }, = 6, I = I nd ( I) = 6I}; 1 16} be semgroup neutrosophc complex modulo nteger lner lgebr over the semgroup S = Z 7. Exmple 3.8: Let V = {( 1,,, 5 ) C ( Z 13 I )= { + b + ci + di, b, c, d Z 13 }, = 1, I = I nd ( I) = 1I} 1 5} be semgroup complex neutrosophc modulo nteger strong lner lgebr over the semgroup S = Z 13. We hve three types of substructures ssocted wth semgroup neutrosophc complex modulo nteger strong lner lgebr. Exmple 3.81: Let 1 3 V = C ( Z 19 I ) = { + b + ci + di, b, c, d Z 19 }, = 18, I = I nd ( I) = 18I}; 1 9} be semgroup complex neutrosophc modulo nteger strong lner lgebr over the semgroup S = Z 19 I. 1 3 W = 5 6 C ( Z 19 I ) ; 1 6} be semgroup complex neutrosophc modulo nteger strong lner sublgebr of V over the semgroup S. Tke M = 1 3 C ( Z 19 I ) ; 1 3}, semgroup complex neutrosophc modulo nteger strong lner sublgebr of V over the subsemgroup T = Z 19 S. 167

169 1 B = 3 C ( Z 19 I ); 1 3} V s only pseudo semgroup complex neutrosophc modulo nteger strong lner sublgebr of V over S. Clerly B s not closed wth respect to product s 1 x y 1x + z 1y 3 z = 3x 3y B where s 1 x y 3 nd z re n B. Hence B s only semgroup complex neutrosophc modulo nteger lner sublgebr over the semgroup whch s not strong lner sublgebr over the semgroup S. Tke 1 3 P =,, 1,, 3 C ( Z 19 I )} V, P s only pseudo semgroup neutrosophc complex modulo nteger pseudo vector subspce of V over S. Clerly V s not closed wth respect to ddton or multplcton. Hence we see we cn defne severl types of substructures n cse of strong lner lgebrs defned over semgroups. Exmple 3.8: Let V =,( 1,,..., ),

170 C ( Z 7 I ) = { + b + ci + di, b, c, d Z 7 }, = 6, I = I nd ( I) = 6I}} be semgroup neutrosophc complex modulo nteger vector spce over the semgroup S = Z 7. Clerly V s not semgroup complex neutrosophc modulo nteger lner sublgebr or V s not semgroup complex neutrosophc modulo nteger strong lner sublgebr. Exmple 3.83: Let V = C ( Z 17 I ) = { + b + ci + di, b, c, d Z 17 }, = 16, I = I nd ( I) = 16I}, 1 1} be semgroup neutrosophc complex modulo nteger lner lgebr over Z 17 the semgroup. Clerly V s not semgroup neutrosophc complex modulo nteger strong lner lgebr over the semgroup Z 17. But consder 1 3 M = C ( Z 17 I ) 5 6 = { + b + ci + di, b, c, d Z 17 }, = 16, I = I nd ( I) = 16I}, 1 6} V s only semgroup complex neutrosophc modulo nteger lner sublgebr of V over the semgroup S = Z 17. We cn defne lner trnsformton of two semgroup neutrosophc complex lner lgebr only they re defned over the sme semgroup S. We gve exmples of them. 169

171 Exmple 3.8: Let 1 3 V = 5 6 C ( Z 9 I ) = { + b + ci + di, b, c, d Z 9 }, = 8, I = I nd ( I) = 8I}} be semgroup neutrosophc complex modulo nteger strong lner lgebr over Z 9. W = {( 1,, 3,, 5, 6, 7, 8, 9 ) C ( Z 9 I ) = { + b + ci + di, b, c, d Z 9 }, = 8, I = I nd ( I) = 8I}1 9} be semgroup complex neutrosophc modulo nteger strong lner lgebr over S = Z 9. Defne T : V W where 1 3 T 5 6 = ( 1,, 3,, 5, 6, 7, 8, 9 ); T s lner trnsformton from V to W. Let η : V V be defned by η 5 6 = η s lner opertor on V. Now we cn derve severl relted propertes wth some smple pproprte chnges. Now we proceed onto defne the noton of group neutrosophc complex modulo nteger vector spce /lner lgebr / strong lner lgebr over group G. If V s set wth zero of complex neutrosophc modulo ntegers nd G to be group of ntegers ddton. We cll V group neutrosophc complex modulo nteger vector spce over the group G f ) for every v V nd g G nd gv nd vg re n V. ).v = for every v V nd the ddtve dentty of G. 17

172 We gve exmples of them. Exmple 3.85: Let V = ,,( 1,,..., ), C ( Z 7 I )= { + b + ci + di, b, c, d Z 7 }, = 6, I = I nd ( I) = 6I}; 1 } be group complex neutrosophc modulo nteger vector spce over the group G = Z 7 under ddton. Exmple 3.86: Let M = C ( Z 9 I ) = { + b + ci + di, b, c, d Z 9 }, = 8, I = I nd ( I) = 8I}; 1 3} be group neutrosophc complex modulo nteger lner lgebr over the semgroup G = Z 9. Clerly M s lso group complex neutrosophc modulo nteger vector spce over the group. However V n exmple 3.78 s not lner lgebr only vector spce. Exmple 3.87: Let V = C ( Z 3 I ) 171

173 = { + b + ci + di, b, c, d Z 3 }, =, I = I nd ( I) = I}; 1 16} be group neutrosophc complex modulo nteger strong lner lgebr over the group G = Z 3. Exmple 3.88: Let V = ,( 1,,..., 1), C ( Z 11 I ) = { + b + ci + di, b, c, d Z 11 }, = 1, I = I nd ( I) = 1I}; 1 3} be group neutrosophc complex modulo nteger vector spce over the group G = Z 11. V s not group neutrosophc complex modulo nteger lner lgebr over G = Z 11. Clerly V s not group neutrosophc complex modulo nteger strong lner lgebr over the group G = Z 11. Exmple 3.89: Let V = C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3 }, =, I = I nd ( I) = I}; 1 15} be group neutrosophc complex modulo nteger lner lgebr over the group G = Z 3. Clerly T s not group complex neutrosophc modulo strong lner lgebr over G. 17

174 Exmple 3.9: Let V = 3 8 C ( Z 7 I ) = { + b + ci + di, b, c, d Z 7 }, = 6, I = I nd ( I) = 6I}; 1 1} nd P = C ( Z 7 I ) = { + b + ci + Id, b, c, d Z 7, = 6, ( I) = 6I, I = I}; 1 8} be group complex neutrosophc modulo nteger lner lgebr over the group G = (Z 7, +). Defne η : M P η = η s lner trnsformton from M to P. If T : M M such tht T then T s lner opertor on M. =

175 We cn derve lmost ll propertes for group neutrosophc complex modulo nteger (strong) lner lgebr defned over the group G. We cn lso fnd Hom G (V, W) nd Hom G (V, V) study the lgebrc structure enjoyed by them. Also study the substructure nd wrtng them s drect sum nd pseudo drect sum cn be tken s routne exercse by the nterested reder. 17

176 Chpter our APPLICATIONS O COMPLEX NEUTROSOPHIC NUMBERS AND ALGEBRAIC STRUCTURES USING THEM In ths chpter we study the probble pplctons of complex neutrosophc rels nd complex neutrosophc modulo ntegers. It s pertnent to keep on record such study s very dormnt; we cn sy solvng equtons nd fndng solutons n C( R I ) = { + b + ci + Id, b, c, d R; = 1} hs not been crred out. Clerly ths set C( R I ) contn R nd the lgebrclly closed feld nmely the feld of complex number C = { + b where, b R} s proper subsets. Thus we cn sy ths extended lke feld wll lso gve the roots when the roots re ndermntes. We cn denote the complex neutrosophc number by the - tuple (, b, c, d) the frst coordnte represents the rel vlue, the second coordnte the complex coeffcents, the thrd coordnte the neutrosophc coeffcent nd forth coordnte the complex neutrosophc coeffcent nd (, b, c, d) = + b + ci + di. 175

177 Study of the egen vlues s complex or ndetermnte stnds s one of the pplctons. urther the noton of complex modulo ntegers C(Z n ) s tself very new. Here the complex number s defned s = n 1 when Z n s tken nto ccount. As n vres the vlue of the fnte squre of the complex number lso vry. Hence C(Z n ) = { + b, b Z n nd = n 1}. Thus C(Z 3 ) ={ + b, b Z 3 nd = }. Now n due course of tme these new structures wll fnd very mny pplctons. nlly the noton of complex neutrosophc modulo ntegers s defned. Ths s lso represented s -tuple wth specl vlue for the fnte complex number, s defned s = n 1 nd (I ) = (n 1)I. Thus C( Z n I )={ + b + ci + di, b, c, d Z n ; = n 1, I = I nd ( I) = (n 1)I}. These new structures re gven lgebrc structures lke groups semgroups, rngs, vector spces nd lner lgebrs nd they wll fnd pplcton n due course of tme once ths reserch becomes populr. 176

178 Chpter ve SUGGESTED PROBLEMS In ths chpter we suggest over 15 problems some t reserch level nd some just routne exercses. 1. Obtn some nterestng propertes enjoyed by () neutrosophc complex rels. (b) neutrosophc complex modulo ntegers. (c) neutrosophc complex rtonls.. Cn ny geometrcl nterpretton be gven to the feld of neutrosophc complex numbers C ( Q I )? 3. Cn C ( Z I ) be Smrndche rng?. Is {( 1, ) 1, C( Z I ) under product } Smrndche semgroup? 177

179 1 5. Let V = 3 C( Q I ), = 1,, 3,, +} be group. ) Defne utomorphsm η : V V so tht ker η s nontrvl subgroup. ) Is V C( Q I ) C( Q I ) C( Q I ) C( Q I )? 6. Let M = 1 3 C( Q I ), 1 } be semgroup under multplcton. ) Prove M s S-semgroup. ) Is M commuttve? ) nd t lest three zero dvsors n M. v) Does M hve dels? v) Gve subsemgroups n M whch re not dels Let S = C( Q I ), 1 1} be neutrosophc complex rtonl semgroup under +. ) nd subsemgroups of S. ) Cn S hve dels? ) Cn S hve dempotents? v) Cn S hve zero dvsors? Justfy your clm Let V = C( Q I ), 1 } be 3 semgroup of neutrosophc complex rtonls under product. 178

180 ) Is V commuttve? ) Cn V hve dempotents? ) Does V hve subsemgroup whch s not n del? v) Gve n del n V. v) Cn V hve zero dvsors? v) Is V Smrndche semgroup? v) Is V Smrndche commuttve semgroup? 9. Let V = C( Q I ), 1 } be complex neutrosophc group under ddton. ) Is V commuttve? b) nd subgroups of V. c) nd subgroup H nd descrbe V /H. d) Defne η : V V so tht... ker η.... e) nd η : V V so tht η -1 exsts. 1. Prove C( Z I ) s group under ddton nd only semgroup under multplcton. Is C ( Z I ) n ntegrl domn? Justfy your clm. 11. Is C( Q I ) = { + b + ci + di, b, c, d Q} feld? Is C( Q I ) prme feld? 1. Cn one sy for ll polynomls wth complex neutrosophc coeffcents C ( R I ) s the lgebrclly closed feld? 13. Cn C( R I ) [x] hve rreducble polynomls? 1. nd rreducble polynomls n C( Q I )? 15. nd rreducble polynomls n C( Z I )? Is every del n C( Z I ) prncpl? Justfy your clm. 179

181 b 16. Study the rng P = c d, b, c, d, C( Z I )}. 17. Is G = b c d d bc,, b, c, d, C( Z I )} group? Is G smple? 18. nd zero dvsors n P mentoned n problem (16) 19. Wht re the dvntges of usng the lgebrc structure C( R I )?. Gve some uses of ths complete lgebrc structure C( R I ). 1. Wht s the crdnlty of the semgroup S = b, c, d, C ( Z I ), }? b c d, b c. nd the number of elements n T = d e f, b, c, d, e, f C( Z 3 I )}, the semgroup under ddton. b 3. Is M = c d, b, c, d, C( Z 5 I )} rng?. ) nd dels f ny n M. b) Is M S-rng? c) Wht s the order of M? d) Cn M hve S-zero dvsors? e) Does M contn dempotents?. Is C( Z 19 I ) feld? 5. Prove C( Z 5 I ) cn only be rng. 18

182 6. Cn C( Z 1 I ) be S-rng? Justfy. 7. Cn C( Z 5 I ) hve S-dels? 8. nd dels n C( Z 6 I ). 9. nd mxmum dels of C( Z 18 I ). 3. nd S-zero dvsors nd S-unts f ny n C( Z I ). 31. Is C( Z 11 I ) Smrndche semgroup under product? 3. Does S = C( Z 15 I ) hve S-dels where S s semgroup under? 33. Let R = C( Z I ) be rng. ) nd S-subrngs of R. ) Cn R hve S-dels? ) Does R contn S-subrngs whch re not dels? v) nd zero dvsors n R. v) Is every zero dvsor n R S-zero dvsor? v) nd S-dempotent f ny n R. v) Determne the number of elements n R. v) Is R S-rng? x) nd n del I n R so tht R / I s feld. x) Does there exst n del J n R so tht R/J s S-rng? 3. Let V = {(x 1, x, x 3, x, x 5 ) x C( Z 11 I ); 1 5} be vector spce over Z 11. ) nd dmenson of V. ) Is V fnte dmensonl? ) nd bss for V. v) nd subspces of V. v) nd Hom z11 (V, V). v) nd lner opertor T on V so tht T -1 does not exst. v) Wrte V s drect sum. 181

183 v) Wrte V s pseudo drect sum. x) Defne T : V V nd verfy Null T + Rnge T = dm V. 35. Let V = C( Z I ); 1 1} be set complex neutrosophc lner lgebr over the set S = 3Z 5Z. ) nd bss of V. ) Is V fnte dmensonl? ) Wrte V s drect sum. v) Wrte V s pseudo drect sum. v) Obtn condtons on lner opertor T of V so tht T -1 exsts. v) Cn V hve subset complex neutrosophc lner sublgebrs? v) nd the lgebrc structure enjoyed Hom S (V,V). v) Cn V be double lner lgebr? 36. Let V = , ( 1,, 3 ), , 5 x be set neutrosophc complex vector spce wth = C( Q I ), over the set S = 7Z 3Z. ) nd bss for V. ) Is V fnte dmensonl? ) Wrte V s drect sum of subspces. v) nd lner opertor on T so tht T -1 does not exst. v) Does V contn subset vector subspce? v) Cn V be wrtten s pseudo drect sum? v) Let V = W 1 W fnd projecton E on V nd descrbe ther propertes. 18

184 Let V = , x,, = C ( Q I ), 1} be specl set neutrosophc complex vector spce over the set S = ZI 3Z 5Z C (Z). ) nd bss. ) Wht s the dmenson of V over S? ) Does there exsts subset vector subspce W of dmenson less thn fve over subset T of S? v) Wrte V s drect sum of set subspces. v) nd non nvertble lner opertor on V. v) Wrte V s pseudo drect sum nd defne projectons. Is tht possble? 38. Let V = 1 3 C( Q I ), 1 } be semgroup neutrosophc complex double lner lgebr over S = Z. ) nd bss of V over Z ) Wht s the dmenson of V over Z? ) If Z s replced by C (Z) wht s dmenson of V? v) If Z s replced by Z I wht s dmenson of V? v) If Z s replced by C( Z I ) wht s dmenson of V? v) Is Z s replced by C(Q) wht s dmenson of V? v) If Z s replced by Q wht s dmenson of V over Q? v) If Z s replced by Q I wht s dmenson of V over Q I? x) If Z s replced by C( Q I ) wht s dmenson of V over C( Q I )? Compre the dmenson n () to (v) nd derve conclusons bsed on t. 183

185 39. Let V = C ( Z I ), 1 9} be group complex neutrosophc double lner lgebr over the group Z. ) nd bss of V. ) Does V hve complex neutrosophc pseudo double lner sublgebrs?. Let V = C ( Z 3 I ) be semgroup under multplcton. ) Is V S-semgroup? ) nd order of V. ) nd zero dvsors f ny n V. v) Cn V hve S-zero dvsors? v) Cn V be group? v) Cn V hve S-unts? v) Cn V hve S-subsemgroups? 1. Let M = { + b + ci + di, b, c, d Z 6 } be semgroup under multplcton. Study questons () to (v) suggested n problem for V.. Prove G = C ( Z 8 I ) cn be group under ddton nd only semgroup under multplcton. nd order of G. 3. Let G = C ( Z 7 I ); ) Is G group? ) Cn G be rng? ) Cn G be feld? v) Wht s the strongest lgebrc structure enjoyed by G?. Let R = C ( Z 1 I ) be complex neutrosophc rng. ) nd the order of R. ) Is R S-rng? ) Cn R hve S-subrngs? v) Cn R hve subrngs whch re not S-dels? v) Cn R hve S-dempotents? 18

186 v) Does R contn zero dvsors whch re not S-zero dvsors? v) Defne η:r R so tht I = ker η s proper subrng of R; so tht R/I s feld. v) Cn R hve Z 1 s ts subrng? x) Is C(Z 1 ) subrng of R? x) Cn R hve S-unts? x) Does there exst n del I n R so tht R/I s feld? 5. Let V = C( Z 3 I ) = { + b + ci + di, b, c, d Z 3 } be complex neutrosophc vector spce over the feld Z 3. ) nd bss for V. ) Wrte V s drect sum. ) Wrte V s pseudo drect sum. v) nd subspces of V so tht V s nether drect sum nor pseudo drect sum. v) nd the lgebrc structure enjoyed by Hom z (V,V). 3 v) Wht s the lgebrc structure of L (V, Z 3 )? 6. nd some nterestng propertes of set complex neutrosophc vector spce defned over the set S. 7. Study the specl propertes ssocted wth the neutrosophc complex modulo nteger rng C( Z 9 I ). 8. Wht re the dstnct propertes of the complex modulo nteger C(Z n ); n not prme? 9. Enumerte the propertes of the complex modulo nteger C(Z p ); p prme. 5. nd the zero dvsors nd unts of C(Z ). 51. nd n del I n C(Z 18 ) so tht C(Z 18 )/I s feld. 5. Does there exst n del I n C(Z 9 ) so tht C(Z 9 )/I s feld? 53. nd subrng S n C(Z n ) so tht S s not n del. 185

187 5. Is every complex modulo nteger rng S-rng? 55. nd necessry nd suffcent condton for complex modulo nteger rng S = C(Z n ) to hve dels I such tht the quotent rng s never feld. 56. Does every C(Z n ) contn zero dvsor? Justfy your clm. 57. Cn every C(Z n ) be feld? 58. Is every element n C(Z 7 ) nvertble? 59. Let G = C(Z ) = { + b, b Z, = 1}; 1 1} be complex modulo nteger group under ddton. ) nd the order of G? ) nd three subgroups nd verfy Lgrnges theorem for G. ) Does G hve p-sylow subgroup? v) Does G stsfy Cuchy theorem for fnte beln groups. v) Wht s the order of A = where C(Z )}? 6. Let G = x C(Z 3) be set. = ) Cn G be group under ddton? ) Wll G be group under multplcton? ) Wll G be semgroup of complex modulo ntegers under multplcton? v) Cn G hve norml subgroup under +? 186

188 v) Is G torson free subgroup of torson group? v) Cn G hve subgroups of fnte order under +? Let S = C(Z 31 ) = { + b, b Z 31, = 3}} be complex modulo nteger semgroup under mtrx multplcton. ) nd order of S. ) Is S commuttve? ) nd dels n S. v) Is S S-semgroup? v) Cn S hve subsemgroups whch re not dels? v) Cn S hve zero dvsors? v) Cn S hve dempotents? v) Gve S-dels n ny n S. x) Cn S hve nlpotent elements? 6. Let S = C(Z 7 ) = { + b, b Z 7, nteger vector spce over the feld Z 7. ) nd dmenson of V over Z 7. ) nd the order of V. ) Wrte V s drect sum. v) Wrte V s pseudo drect sum over Z 7. v) Defne lner opertor T on V such tht W = = 6}; 1 3} be complex modulo , j C(Z 7 ); 1 6, j=1,, 3} V so tht T (W) W. v) nd S on V so tht S (W) W. 187

189 Let V = C (Z ) = { + b, b Z, = 19}; 1 9} be complex modulo nteger group under ddton. ) nd order of V. ) Verfy Lgrnge s theorem. ) Wht re the orders of subgroups of V? v) nd quotent groups v) nd order of x = V v) Is Cuchy theorem true for x n V? Let V = C (Z 3 ) = { + b , b Z 3, = 9}; 1 16} be Smrndche complex modulo nteger lner lgebr over the S-rng Z 3. ) nd bss of V. ) Is V fnte dmensonl? ) Is order of V fnte? v) Wrte V s drect sum. v) Wrte V s pseudo drect sum. v) nd sublner lgebr W of V such tht for lner opertor T, T(W) W. v) nd pseudo S-subvector spce of V over Z

190 Let S = 1 C (Z 3 ) = { + b, b Z 3, = }; 1 } be complex modulo nteger group under ddton. ) nd subgroups of S. ) Cn S be wrtten s ntervl drect sum of complex modulo nteger subgroups? ) Is every element n S s of fnte order? v) Wht s the order of S? v) Cn S hve p-sylow subgroups? v) nd ll the p-sylow subgroups of G. 66. Let G = 1 3 C(Z ) = { + b, b Z, = 39}; 1 } be semgroup of complex modulo ntegers under mtrx multplcton. ) Is G commuttve? (Prove your clm) ) Wht s the order of G? ) Cn G hve S-subsemgroups? v) Is G S-semgroup? v) Cn G hve dels? v) Cn G hve S-dempotents? 1 v) Let H = 3 1 3, C(Z ); 1 } G. Is H subgroup of G. v) Wht s the order of H? x) Is H commuttve structure wth respect to product? x) Cn G hve zero dvsors whch re not S-zero dvsors? x) Cn G be wrtten s drect sum of subsemgroups? 189

191 67. Let S = 1 3 C(Z ) = { + b, b Z, = 19}; 1 3} be complex modulo nteger rng of chrcterstc. ) Is S commuttve rng? ) Is S S-rng.? ) Cn S hve S-dels? v) Cn S hve dels whch re not S-dels? v) Wht s the order of S? v) Cn S hve subrngs whch re not S-subrngs? v) Does S contn mxml del? v) Cn S hve zero dvsors? x) Cn S hve S-unts? x) Does S contn subrngs whch re not S-dels? x) nd homomorphsm η : S S so tht ker η. nd S/ker η. 68. Let M = x C(Z 5 ) = { + b, b Z 5, = }} be complex modulo nteger polynoml rng. ) Is M feld? ) Cn M be S-rng? ) Is M n ntegrl domn? v) Is M prncpl del domn? v) Cn M hve rreducble polynomls? v) Is the polynoml p (x) = (3 + ) + (1+ )x + (3 + )x 3 + x n M reducble polynoml order C(Z 5 )? 69. Let N = = x C (Z 1 ) = { + b, b Z 1, = 1}} be complex modulo nteger polynoml rng. ) Is N feld or n ntegrl domn? ) Is N S-rng? ) Cn N hve zero dvsors? = 19

192 v) Is N prncpl del domn? v) Cn N hve S-subrngs? v) Does N contn S-unts? v) Cn N hve subrngs whch re not dels? v) Cn N hve S-dempotents? x) Cn N hve dels whch re mnml? 7. Let V = x C (Z 16 ) = { + b, b Z 16, = = 15}} be complex modulo nteger polynoml rng. Answer the problem (1) to (x) of problem (69) for V. 71. Let W = (C (Z 1 ) C (Z 1 ) C (Z 3 )) = { + b, c+d, e+m ), b Z 1, = 11, c, d Z 1, = 9 nd e, m Z 3 ; = }. ) Is W group of complex modulo ntegers under ddton? ) Wht s the order of W? ) Is W semgroup under product? v) Is W s semgroup of S-semgroup? v) nd zero dvsors n W? v) nd ll p-sylow subgroups of W treted s group of complex modulo ntegers? (Is t possble?) v) Cn W be gven complex modulo ntegers rng structure? 7. Let R = (Z 3 C (Z 5 ) C (Z 7 ) Z 11 ) = {(, b, c, d) Z 3, b C (Z 5 ) = {x + y x, y Z 5, = } c C (Z 7 ) = {m + n m, n Z 7, = 6}, d Z 11 } be rng of modulo complex ntegers. ) nd order of R. ) Cn R hve dels? ) Is R S-rng? v) Cn R hve S-unts? v) Is R prncpl del domn? v) Cn R hve S-zero dvsors? v) nd dels n R whch re not S-del. 191

193 v) nd n del I n R nd determne R /I. x) Is t possble to hve R/I to be feld? x) Cn R ever be n ntegrl domn? 73. Let P = {(Z 7 C(Z ) C(Z 11 )) = (, b, c) where Z 7, b C(Z ) = {d + g d, g Z, = 39}, c C(Z 11 ) = {m +n m, n Z 11 ; = 1}} be group under ddton. ) Menton some specl propertes enjoyed by P. ) Wht s the order of P? ) Is P beln? v) nd p-sylow subgroups of P. v) nd subgroups of P whch re only modulo nteger subgroups nd not complex modulo nteger subgroups. 7. Let S = {(Z C(Z 1 ) C(Z 15 ) Z 6 ) = (, b, c, d) Z, b C(Z 1 ) = {e + g e, g Z 1, = 11}, b C(Z 1 ) = {e + g e, g Z 1 ; = 11}, c C(Z 15 ) = {m + n m, n Z 15 ; = 1}, d Z 6 } be complex modulo nteger semgroup under product. ) nd order of S. ) Is S S-semgroup? ) Cn S hve zero dvsors? v) Cn S hve S-unts? v) nd dels n S. v) nd S-subsemgroups whch re not dels n S. v) Does S contn dempotents? v) Cn S hve S-Lgrnge subgroups? 75. Let T = {(C(Z 3 ) C(Z ) C(Z 6 )) = (, b, c) C(Z 3 ) = {x + y x, y Z 3, = }, b C(Z ) = {m + n m, n Z ; = 1}, c C(Z 6 ) = {t + u t, u Z 6 ; = 5} be the rng of complex modulo ntegers. ) Wht s the order of T? ) Is T S-rng? ) nd S-dels f ny n T. 19

194 v) nd subrng n T whch s not n del. v) Is T prncpl del domn? v) Cn T be unque fctorzton domn? v) Does T contn S-zero dvsors? 76. Let M = {(C(Z 3 ) C(Z ) C(Z 5 ) C(Z 6 ) C(Z 7 )) = (, b, c, d, e) C(Z 3 ) = {g + h g, h Z 3, = }, b C(Z ) = {m + n m, n Z ; {t + u t, u Z 5 ; = 3}, c C (Z 5 ) = = }, d C(Z 6 ) = {r + s r, s Z 6 ; = 5}, e C (Z 7 ) = {p + q p, q Z 7 ; = 6}} be group of complex modulo ntegers under ddton. ) nd order of M. ) nd subgroups of M. ) nd ll p-sylow subgroups of M. v) Prove every element n M s of fnte order. v) nd n utomorphsm on M. 77. Cn we hve unque fctorzton domn of complex modulo ntegers? Justfy your clm. 78. Gve n exmple of complex modulo ntegers whch s prncpl del domn (Does t exst!). 79. Wht s the lgebrc structure enjoyed by the complex modulo rng C(Z 19 )? 8. Obtn some nterestng propertes bout complex modulo nteger rngs. 81. Obtn some unque propertes enjoyed by rngs bult usng the complex modulo ntegers C(Z n ). 8. Is C( Z I ) unque fctorzton domn? 83. Cn C( R I ) be prncpl del domn? 193

195 b 8. If A = c d, b, c, d C( Q I ); d bc } be rng of neutrosophc complex modulo ntegers. ) Is A commuttve? ) Is A S-rng? ) Cn A hve S-dels? v) Cn A hve zero dvsors? v) Cn A hve S- subrngs whch re not dels? v) Cn the concept of.c.c. or d.c.c. to mposed on A? 85. Let M = x C( Z I ) be the neutrosophc = complex polynoml rng. ) nd subrngs n M whch re not dels? ) Is M S-rng? ) Study the concept of rreducble nd reducble polynomls n M. v) nd lnerly reducble polynoml n M. v) nd rreducble polynoml of degree three n M. v) Wll n rreducble polynoml n M generte mxml del? v) Is M prncpl del domn? v) Is M unque fctorzton domn? 86. Let R = x C( Q I ) be neutrosophc = complex polynoml rng. Answer the questons () to (v) mentoned n problem (85). 87. Let = x C be rng. nd the dfferences = between M n problem 8, R n problem (83) nd n ths problem. 19

196 88. Suppose P = x C( R I ) be the rel = neutrosophc rng of polynomls. ) Study / nswer questons () to (v). ) Compre P wth R, nd M n problems (8) to (8). 89. Menton the propertes dstnctly ssocted wth complex modulo nteger vector spces defned over Z p, p prme. 9. nd nterestng propertes enjoyed by Smrndche complex modulo nteger lner lgebrs defned over the S-rng Z n, n not prme Let P = C(Z ) = { + b, b Z, = 3}, 1 8} be Smrndche complex nteger vector spce over the S-rng Z. ) nd bss for P over Z. ) Wht s the dmenson of P over Z? ) nd subspces of P. v) Wrte P s drect sum. v) Wrte P s pseudo drect sum. v) nd lner opertor T on P so tht T -1 exsts. v) Does P contn pseudo S-complex modulo nteger vector subspces over feld Z? 9. Let V = x C( Z 7 I ) = { + b + ci + Id, = b, c, d Z 7, = 6, I = I ( I) = 6I}} be the neutrosophc complex polynoml rng. ) Is V S-rng? ) Cn V hve reducble polynomls? 195

197 ) Gve exmples of rreducble polynomls? v) Is V n ntegrl domn? v) Cn V hve zero dvsors? v) Is V prncpl del domn? v) Cn V hve subrngs whch re not dels? 93. Let W = x C( Z 6 I ) = { + b, b Z 6, = = 5}} be neutrosophc complex modulo nteger rng. ) nd zero dvsors n W. ) nd two reltvely prme polynomls n W. ) nd dels n W. v) Is W S-rng? v) nd subrngs n W whch re not dels? v) Cn W hve S-unts? v) Cn W hve S-zero dvsors? v) Is W prncpl del rng? x) Cn W hve mxml dels? x) Does W contn mnml dels? x) Wll.c.c. condton on dels be true n W? x) Does W contn pure neutrosophc dels? x) Cn W hve complex modulo nteger dels? 9. Let G = x C( Z I ) = { + b + ci + di =, b, c, d Z, = 1}} be group under ddton. ) nd order of G. ) Cn G hve Sylow subgroups? ) Cn G be wrtten s drect product of subgroups? v) Let H = 1 x C( Z I ) G. nd G / H. = v) Wht s the order of G / H? 19 v) nd x H. = 196

198 95. Let = 1 3 C( Z 1 I ) = { + b, b Z 1, = 1}, 1 } be rng. ) Is S-rng? ) Is s of fnte order? ) nd subrngs n whch re not dels. v) Is commuttve? v) nd left del n whch s not rght del n. v) nd S-zero dvsors f ny n. v) Cn be rng wth S-unts? v) Wht s the dfference between nd 1 H = Z1 3? x) nd some specl propertes enjoyed by nd not by b T = c d, b, c, d C (Z 1 ) = { + b, b Z 1, = 11}}. b 96. Let P = c d, b, c, d C ( Z 9 I ) = { + b + ci + di, b, c, d Z 9, = 8, I = I, ( I) = 8I}} be complex neutrosophc modulo nteger semgroup under multplcton. ) nd order of P. ) Is P commuttve? ) nd zero dvsors f ny n P. v) Is P S-semgroup? v) Cn P hve S-subsemgroups? v) nd rght dels whch re not left dels n P. v) nd S-unts f ny n P. 197

199 b 97. Let M = c d, b, c, d C ( Z 11 I ) = {m + n + ti + si m, n, t, s Z 11, = 1, I = I, ( I) = 1I}; d bc } be group under multplcton. ) nd order of M. ) Prove M s non commuttve. ) nd norml subgroups n M. v) nd subgroups whch re not norml n M. v) Is Cuchy theorem true n M? v) Show M cn be embedded n symmetrc group S n, gve tht n? (Culey s theorem). 98. nd some nterestng propertes ssocted wth neutrosophc complex modulo nteger semgroups. 99. Let S = {( 1,, 3,, 5 ) C ( Z I ) = { + b + ci + di, b, c, d Z, = 3, I = I, ( I) = 3I}; 1 5} be complex neutrosophc modulo nteger semgroup under multplcton. ) nd order of S. ) nd zero dvsors n S. ) nd dels n S. v) Is S S-rng? v) nd S-subsemrngs n S. 1. Suppose S n problem (98) s tken s ddton wht re the relevnt dfferences you cn fnd? x1 x 11. Let V = x C ( Z I ) = { + b, b Z, x1 = 3, I = I, ( I) = 3I}; 1 1} be group of complex neutrosophc modulo ntegers under ddton. ) nd subgroups. 198

200 ) nd order of V. ) nd p-sylow subgroups of V. v) Wrte V s drect sum. 1. Let P = {(C ( Z 7 I ) C ( Z 11 I ) C ( Z 3 I )) = (, b, c) C ( Z 7 I ) = {d + g + hi + ki d, g, h, k Z 7, = 6, I = I, ( I) = 6I}} b C ( Z 11 I ) = {m + n + ti + si m, n, t, s Z 11, = 1, I = I, ( I) = 1I} nd C C ( Z 3 I ) = {( + b + ei + di) =, I = I, ( I) = I}} be neutrosophc complex modulo nteger semgroup under product. ) nd order of P. ) nd del of P. ) Is P S-semgroup? v) Is T = T ={Z 7 Z 11 Z 3 } P pseudo modulo nteger subsemgroup of P? Cn T be n del of P? v) Cn W = {C(Z 7 ) C (Z 11 ) C (Z 3 )} P be pseudo complex del of P? v) Cn B = { Z 7 I C ( Z 11 I ) C ( Z 3 I )} P be pseudo neutrosophc del of P? 13. Suppose P n problem 11 s tken s neutrosophc complex modulo nteger group under ddton then study the bsc propertes ssocted wth P. Compre P s group under + nd semgroup under multplcton. 1. Let M = {Z 7 C (Z 5 ) C ( Z 1 I ) ( Z 1 I ) = {(, b, c, d) Z 7, b C (Z 5 ) = + p ;, p Z 5, = }; c C ( Z 1 I ) = { + b + di + e I, b, d, e Z 1, = 11, I = I, ( I) = 11I}, d Z 1 I = { + bi, b Z 1, I = I}} be semgroup under multplcton. ) nd order of M. ) nd zero dvsors n M. ) Is M S-semgroup? v) nd S-dels f ny n M. v) Cn M hve S-unts? 199

201 15. Let L = C (Z ) = { + b, b Z, = 39}; 1 7} be complex modulo nteger group under ddton. ) nd ll p-sylow subgroups of L. ) Cn L be wrtten s drect unon sum of subgroups? 1... ) If H = C (Z ); 1 6} L. Is H subgroup? If so fnd the coset of H n L Let M = C ( Z 1 I ) = { + b + ci + di, b,c,d Z 1, = 11}, I = I; (I ) = 11I}; 1 9} be neutrosophc complex modulo nteger rng. ) Is M S-rng? ) Wrte S-unts n M. ) nd dels n M. v) nd S-dels f ny n M. v) nd subrngs whch re not dels. v) Wht s the order of M? v) Does M contn S-subrngs whch re not S-dels? v) nd rght del whch s not left del of M Let T = C ( Z I ) = { b + ci + di, b, c, d Z, = 1, I = I; ( I) = 1I}; 1 16} be neutrosophc complex modulo nteger semgroup.

202 ) nd order of T. ) nd dels n T. ) nd subrngs n T whch re not dels of T. v) nd rght del of T nd left del of T. v) nd S-zero dvsors f ny n T. v) Is T S-semgroup? v) Cn T hve dempotents? 18. Let B = {(x 1, x, x 3, x, x 5, x 6 ) x C ( Z 1 I ) = { + b + ci + di, b, c, d Z 1, = 11, I = I; ( I) = 11I}; 1 6} be rng of neutrosophc complex modulo ntegers. ) nd order of B. ) nd dels of B. ) Cn B hve subrngs whch re not dels? v) Cn B hve S-dels? v) Is B S-rng? v) Cn B hve S-subrngs? v) Cn I = {(x 1 x x 3 ) x C ( Z 1 I ); 1 3} B be n del? v) nd B/I. b 19. Let M = c d + di, b, c, d C ( Z 1 I ) = { + b + ci = 13, I = I; ( I) = 13I}} be rng of neutrosophc complex modulo nteger. ) Is M S-rng? ) nd order of M. ) nd S-dels f ny n M. v) Cn M hve S-zero dvsors? v) nd dempotents f ny n M. v) nd left del of M whch s not rght del of M nd vce vers. v) Does M hve zero dvsor whch s not S-zero dvsor? v) nd S-subrng whch s not n del of M. 1

203 11. Let T = 1 x C ( Z 3 I ) = { + b + ci + di =, b, c, d Z 3 ; =, I = I; ( I) = I}} be semgroup (or group under ddton). ) nd order of T. ) Wht re the strckng dfferences when T s consdered s group nd s semgroup? ) nd ll p-sylow subgroups of T. v) nd S n (exct n) so tht T s embeddble n S n. v) Wht re the dstnct fetures s group T enjoys? v) Cn T hve dels, (T s semgroup)? 111. Let M = {ll 1 1 upper trngulr mtrces wth entres from C ( Z 5 I ) = { + b + ci + di, b, c, d Z 5 ; =, I = I; ( I) = I}} be neutrosophc complex modulo nteger rng. ) Is M commuttve? ) Is M S-rng? ) Is M fnte? v) nd subrngs n M whch re not dels? v) nd S-dels f ny. v) Does M hve zero dvsors? v) Cn M hve S-unts? v) nd ny specl property enjoyed by M. 11. Let V = {(x 1, x,, x 1 ) x C ( Z 11 I ) = { + b + ci + di, b, c, d Z 11, = 1, I = I; ( I) = 1I}} be neutrosophc complex modulo nteger lner lgebr over the feld Z 11. ) nd subspces of V. ) Wrte V s drect sum of subspces (lner sublgebrs). ) nd bss of V. v) If V s defned over Z 11 I. fnd the bss. v) If V s defned over C (Z 11 ) wht s the dmenson of V over C (Z 11 )?

204 v) Wht s dmenson of V f V s defned over C ( Z 11 I )? 113. Let M = C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3, =, I = I; ( I) = I}; ; 1 16} be neutrosophc complex modulo nteger vector spce over Z 3. ) nd bss of M over Z 3. ) Wht s dmenson of M over Z 3? ) Wrte M s drect sum. v) Wrte M s pseudo drect sum. 1 3 v) Let W = C ( Z 3 I ) 1 8} M be subspce of M. nd lner opertor on V so tht T (W) W. v) Study (1) nd () f Z 3 f replced by C ( Z 3 I ). 11. Let P = 1 3 C ( Z 1 I ) = { + b + ci + di, b, c, d Z 1, = 11, I = I; ( I) = 11I}; ; 1 } be S-neutrosophc complex modulr nteger lner lgebr over the S-rng Z 1. ) nd bss of P. ) Wht s the dmenson of P over Z 1? ) Wrte P s drect sum. v) Wrte P s pseudo drect sum. v) nd the lgebrc structure enjoyed by Hom Z1 (V,V). v) Gve lner opertor T such tht T -1 exsts nd keeps no subspce of P nvrnt under t. v) Does such lner opertor exst? 3

205 5 1, 1 3, x 3 C = ( Z I ) = { + b + ci + di = 19, I = I; ( I) = 19I}; 5} be set neutrosophc complex nteger modulo vector spce over the set S = {,, 5, 1, 11} Z. ) nd the number of elements n V. ) nd bss of V over S. ) Does V hve subspces? v) Cn V be wrtten s drect sum? If so do t. v) Wrte V s pseudo drect sum Let V = ( ) 116. Let P = x,,, 5 6, = C ( Z 11 I ) = { + b + ci di, b, c, d Z 11, = 1, I = I; ( I) = 1I}; } be set neutrosophc complex vector spce of modulo ntegers over the set S = {, 3, 5, 7} Z 11. ) nd dmenson of P over S. ) nd bss of P over S. ) Wrte P s drect sum.

206 v) nd T : P P such tht T keep tlest two subspces nvrnt. v) Prove P cn hve pure neutrosophc subspces over S. v) nd complex modulo nteger set vector subspces of P over S. v) nd just modulo nteger vector subspces of P over S. v) nd the structure enjoyed by Hom S (P, P). x) Cn lner opertor on P be such tht t keeps every subspce nvrnt? 117. Let M = C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3, =, I = I; ( I) = I}; 1 18} be set neutrosophc complex lner lgebr over the set S = {, 1,, 1, 9, } Z 3. ) nd bss for M over S. ) Wrte M s drect sum of sublner lgebrs over S. ) nd dmenson of M over S. v) nd two dsjont sublner lgebrs of M so tht the ntersecton s the 3 6 zero mtrx. v) nd lner opertor T on M so tht T -1 does not exst Let V = C ( Z 3 I ) = { + b + ci di, b, c, d Z 3, =, I = I; ( I) = I}; 1 15} be set neutrosophc complex modulo nteger lner lgebr over the set S = {, 1,, 19, 1, 3, 5, 7} Z 3. 5

207 ) nd bss for V. ) Wht s the dmenson of V over S? ) Wrte V s drect sum of set sublner lgebrs. v) Study Hom S (V, V). v) Study L (V, S). v) Introduce nd study ny other propertes relted wth V Let G = C ( Z 5 I ) = { + b + ci + di, b, c, d Z 5, =, I = I; ( I) = I}; 1 1} be group of neutrosophc complex modulo ntegers under ddton. ) Wht s the order of G? ) nd ll p-sylow subgroups of G. ) Let H = 1 C( < Z 5 I > );1 3 be subgroup of G. fnd G / H. Wht s o (G/H)? 1. Let P =, b, c, d Z 5, C ( Z 5 I ) = { + b + ci + di =, I = I; ( I) = I} be group under ddton of neutrosophc complex modulo ntegers. ) nd order of P. ) Obtn ll p-sylow subgroups of P. ) Gve subgroup of P whch s not p-sylow subgroup. v) Cn P be group under product? v) Is t possble to wrte P s drect sum of subgroups? v) Cn η : P P be such tht ker η s nontrvl? (η - group homomorphsm). 11. Gve ny nce nd nterestng property enjoyed by neutrosophc complex modulo nteger groups. 6

208 1. Obtn some clsscl theorems of fnte groups n cse of complex modulo neutrosophc nteger groups. 13. Construct clss of neutrosophc complex modulo nteger semgroups whch re not S-semgroups. 1. Gve n exmple of complex neutrosophc modulo nteger semgroup whch s S-semgroup. 15. Does there exst rngs bult usng neutrosophc complex modulo ntegers whch s not S-rng? 16. Study the lgebrc structure of Hom Z (V, V) where V s p neutrosophc complex modulo nteger vector spce over Z p. 17. Study the lgebrc structure of L (V, Z p ) where V s the neutrosophc complex modulo nteger vector spce over Z p. 18. Let V = C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3, =, I = I nd ( I) = I}; 1 3} be neutrosophc complex modulo nteger vector spce over the feld Z 3. nd L (V, Z 3 ). 19. Let V = , ( 1,,, 1 ), 1 7 C ( Z 7 I ) = { + b + ci + d I, b, c, d Z 7, = 6, I = I nd ( I) = 6I}; 1 1} be set complex 7

209 neutrosophc modulo nteger vector spce over the set S = {, I, 3+7I, 7I, 5+I, 1}. ) nd bss. ) Is V fnte or nfnte dmensonl over S? ) Wrte V s drect sum. v) Wrte V s pseudo drect sum. v) nd Hom S (V,V). v) nd tlest one lner opertor T on V so tht T -1 exsts. 13. nd some nce pplctons of set complex neutrosophc modulo nteger vector spces defned over set S Wht s the dvntge of usng set complex neutrosophc modulo nteger strong lner lgebrs over the set S? 13. Let V be set neutrosophc complex modulo nteger strong lner lgebr defned over set S. nd Hom S (V,V) Let M = 1 3 C ( Z 3 I ), 1 } where C ( Z 3 I ) = { + b + ci + di, b, c, d Z 3, =, I = I nd ( I) = I} be group complex neutrosophc modulo nteger strong lner lgebr over the group G = Z 3. ) nd bss of M over G. ) Show the number of elements n M s fnte (nd o (M)). ) Wrte M s pseudo drect sum. v) Is t possble to wrte M s drect sum of strong lner sublgebrs? 13. nd some nterestng propertes ssocted wth group modulo nteger neutrosophc complex vector spces defned over group G. 8

210 135. Compre group neutrosophc complex modulo nteger vector spces wth group neutrosophc complex modulo nteger strong lner lgebrs nd the dfference between the semgroup complex neutrosophc modulo nteger vector spces nd semgroup complex neutrosophc modulo nteger strong lner lgebrs Let V = {( 1,,, 9 ) C ( Z 5 I ) = { + b + ci + di, b, c, d Z 5, =, I = I nd ( I) = I}; 1 9} be semgroup neutrosophc complex modulo nteger strong lner lgebr over the semgroup S = C (Z 5 ) = { + b, b Z 5 } under ddton. ) nd the number of elements n V. ) nd bss of V. ) Wht s the dmenson of V over S? v) Wrte V s drect sum of sublner lgebrs. v) Wrte V s pseudo drect sum nd the lgebrc structure enjoyed by Hom S (V,V); V set complex neutrosophc modulo ntegers over the set S If V s neutrosophc complex modulo nteger lner lgebr over Z p fnd Hom (V, V). Zp Let M = C ( Z I ) = { + b + ci + di, b, c, d Z, = 1, I = I nd ( I) = I}; 1 } be complex neutrosophc modulo nteger vector spce over the feld Z. ) nd dmenson of M over Z. ) Wht s the order of M? ) nd bss of M over Z. v) Is M lner lgebr? 9

211 v) Cn M be strong lner lgebr? v) Wrte M s drect sum of subspces. v) nd Hom Z (M, M). v) nd L (M, Z ). x) Wrte M s pseudo drect sum of subspces. x) If Z s replced by C ( Z I ) wht s the structure of M? Let M = C ( Z 11 I ) = { b + ci + di, b, c, d Z 11, = 1, I = I nd ( I) = 1I}; 1 3} be set complex neutrosophc modulo nteger lner lgebr over the set S = {, 1}. ) nd bss of M. ) nd the number of elements n M. ) Wrte M s drect sum of subspces. v) nd Hom S (M, M). v) nd T Hom S (M, M) such tht T -1 does not exst. 1. If S n problem 1 s replced by Z 11, nswer questons (1) to (v). 13. If S n problem 1 s replced by C (Z 11 ) then study the questons (1) to (v). 1. If S n problem 1 s replced by C ( Z 11 I ), wll S be vector spce? Let V = C ( Z I ) = { + b + ci + di, 5 6 b, c, d Z, = 19, I = I nd ( I) = 19I}; 1 6} 1

212 be Smrndche neutrosophc complex modulo nteger vector spce over the S-rng Z. ) nd bss of V. ) nd number of elements of V. ) Wrte V s drect sum of subspces. v) nd Z Hom (V, V). v) nd projecton opertor on V. v) Does there exsts lner opertor on V whch keeps every subspce of V nvrnt? b 16. Let M = c d, b, c, d C ( Z 1 I ) = {m + n + ri + si m, n, r, s Z 1, = 9, I = I nd ( I) = 9I}} be Smrndche complex neutrosophc modulo nteger lner lgebr over the S-rng S = Z 1. ) nd bss of M. ) nd dmenson of M over S - rng. ) If M s treted only s vector spce wll dmenson of M over S be dfferent? Justfy your clm. v) Cn M be wrtten s drect sum of lner sublgebrs? v) nd the lgebrc structure enjoyed by Hom S (M, M). v) Wrte M s pseudo drect sum of lner sublgebrs. v) Is P =,b C( Z1 I ) b M be lner sublgebr? b c d 17. Let V =, b,c,d,e,f,g, h C(Z 6) e f g h be the Smrndche complex modulo nteger vector spce over the S-rng Z 6. ) nd bss of V over Z 6. ) Is V fnte dmensonl over Z 6? ) nd Hom Z (V, V). 6 v) If Z 6 s replced by C (Z 6 ) wll the dmenson of V 11

213 over C (Z 6 ) vry? Let T = 5 6 C(Z 7 );1 9 be complex modulo nteger lner lgebr over the feld = Z 7. ) nd bss of T over Z 7. ) nd L (T, Z 7 ). ) nd Hom Z (T, T). 7 v) nd the number of elements n T. v) Wrte T s pseudo drect sum of sublner lgebrs. v) Cn T be wrtten s drect sum of sublner lgebrs? 19. Enumerte some nterestng propertes enjoyed by complex modulo nteger vector spce. 15. Wht s the dstnct fetures enjoyed by Smrndche complex modulo nteger vector spces defned over S- rng (Z n ) nd complex modulo nteger vector spces defned over feld Z p Let V = C ( Z 3 I ) = { b + ci + di, b, c, d Z 3, =, I = I nd ( I) = I}; 1 16} nd W = {( 1,,, 1 ) C ( Z 3 I ); 1 1} be two complex neutrosophc modulo nteger lner lgebr defned over the feld Z 3. ) nd bss of V nd bss of W. ) Wht s the dmenson of V? ) nd Hom Z (V, W) = S. 3 v) nd Hom Z (W, V) = R. 3 v) Is R S? 1

214 15. Obtn some nterestng pplctons of complex modulo nteger lner lgebrs Determne some nce pplctons of complex neutrosophc modulo nteger vector spce. 15. nd ny property enjoyed by S - complex neutrosophc modulo nteger vector spces defned over S-rng Let V = C ( Z 9 I ) = { + b + ci + di, b, c, d Z 9, = 8, I = I nd ( I) = 8I}; 1 } be set complex neutrosophc lner lgebr of modulo nteger defned over the set S = {, 1, I}. ) nd bss of V over S. ) Wll the dmenson of V chnge f V s replced by Z 9. ) nd Hom S (V, V) Let P = {( 1,, 5 ) C ( Z 13 I ) = { + b + ci + di, b, c, d Z 13, = 1, I = I nd ( I) = 1I}; 1 5} be set complex neutrosophc modulo nteger strong lner lgebr over S = {, I}. ) nd bss of P over S. ) nd dmenson of P over S. ) Wrte P s drect sum of strong lner sublgebrs. v) If S s replced by T = {, 1} wll dmenson of P over T dfferent? 157. nd pplctons of complex neutrosophc rel lner lgebrs. 13

215 b 158. Let S = c d, b, c, d C ( R I ) = {m + n + si + Ir = 1, I = I nd (I) = I}} be complex neutrosophc lner lgebr over R. ) nd bss of S over R. ) Is S fnte dmensonl over R? ) nd Hom R (S,S). v) nd L (S, R). v) If R s replced by R I study results (1) to (v). v) If R s replced by C (R) = { + b, b R, = 1} = C study questons (1) to (v). v) If R s replced by C ( R I ) Wht s dmenson of S? Study (1) to (v) questons. v) Compre nd dstngush between the spces gven n questons (v) (v) nd (v) Obtn some nterestng results bout complex neutrosophc semvecor spces defned over Z. b 16. Let P = c d, b, c, d C ( Z I ) = {m + n + ri + si = 1, I = I nd (I) = I}, m, n, r, s, re n Z}. Is P semfeld? Justfy your nswer. 1

216 URTHER READINGS 1. Brkhoff, G. nd Brtee, T.C. Modern Appled Algebr, Mc- Grw Hll, New York, (197).. Cstllo J., The Smrndche Semgroup, Interntonl Conference on Combntorl Methods n Mthemtcs, II Meetng of the project 'Algebr, Geometr e Combntor', culdde de Cencs d Unversdde do Porto, Portugl, 9-11 July Chng Qun, Zhng, Inner commuttve rngs, Sctun Dscue Xuebo (Specl ssue), 6, (1989).. Hll, Mrshll, Theory of Groups. The Mcmlln Compny, New York, (1961). 5. Hersten., I.N., Topcs n Algebr, Wley Estern Lmted, (1975). 6. Ivn, Nvn nd Zukermn. H. S., Introducton to number theory, Wley Estern Lmted, (198). 7. Lng, S., Algebr, Addson Wesley, (1967). 8. Rul, Pdll, Smrndche Algebrc Structures, Smrndche Notons Journl, 9, (1998). 9. Smrndche, lorentn, (edtor), Proceedngs of the rst Interntonl Conference on Neutrosophy, Neutrosophc Set, Neutrosophc Probblty nd Sttstcs, Unversty of New Mexco, (1). 15

217 1. Smrndche, lorentn, A Unfyng eld n Logcs: Neutrosophc Logc, Prefce by Chrles Le, Amercn Reserch Press, Rehoboth, 1999,. Second edton of the Proceedngs of the rst Interntonl Conference on Neutrosophy, Neutrosophc Logc, Neutrosophc Set, Neutrosophc Probblty nd Sttstcs, Unversty of New Mexco, Gllup, (1). 11. Smrndche, lorentn, Specl Algebrc Structures, n Collected Ppers, Abddb, Orde, 3, (). 1. Smrndche lorentn, Mult structures nd Mult spces, (1969) Smrndche, lorentn, Defntons Derved from Neutrosophcs, In Proceedngs of the rst Interntonl Conference on Neutrosophy, Neutrosophc Logc, Neutrosophc Set, Neutrosophc Probblty nd Sttstcs, Unversty of New Mexco, Gllup, 1-3 December (1). 1. Smrndche, lorentn, Neutrosophc Logc Generlzton of the Intutonstc uzzy Logc, Specl Sesson on Intutonstc uzzy Sets nd Relted Concepts, Interntonl EUSLAT Conference, Zttu, Germny, 1-1 September Vsnth Kndsmy, W. B., On Smrndche Cosets, (). dels.pdf 16. Vsnth Kndsmy, W. B., Smrndche Semgroups, Amercn Reserch Press, Rehoboth, NM, (). Book1.pdf 17. Vsnth Kndsmy, W. B. nd Smrndche,., N- lgebrc structures nd S-N-lgebrc structures, Hexs, Phoenx, Arzon, (5). 16

218 INDEX C Commuttve fnte complex rng, 86-8 Complex modulo nteger vector spce, 1-3 Complex modulo nteger vector subspce, 1-5 Complex neutrosophc complex subspce, 3-1 Complex neutrosophc modulo nteger lner lgebr, 131- Complex neutrosophc modulo nteger vector spce, 1-5 Complex neutrosophc rtonl lke feld, 1-3 Complex-complex neutrosophc vector spce, 18-9 Complex-complex neutrosophc vector subspce, 18-9 nte complex modulo ntegers, 83-5 uzzy complex neutrosophc semgroup, 79-8 uzzy neutrosophc complex semvector spce, 8- uzzy semgroup neutrosophc complex vector spce, 77-9 G Group complex neutrosophc lner lgebr, 59-6 Group neutrosophc complex double lner lgebr, 73-5 Group neutrosophc complex modulo nteger vector spce, Group neutrosophc complex number vector spce,

219 I Integer complex neutrosophc group, 8-9 Integer complex neutrosophc del, 9-1 Integer complex neutrosophc ntegrl domn, 13-5 Integer complex neutrosophc monod, 8-9 Integer complex neutrosophc n n mtrx rng, 1- Integer complex neutrosophc polynoml rng, 13-5 Integer complex neutrosophc subrng, 9-1 Integer neutrosophc complex commuttve rng, 9-1 Integer neutrosophc complex monod, 8-9 Integer neutrosophc complex number, 7-9 N n n mtrces of complex modulo nteger lner lgebr, Neutrosophc complex modulo ntegers, Neutrosophc complex rtonl lke feld, 1-3 Neutrosophc lke feld of rtonls, 1-3 Neutrosophc lke feld, 19- Neutrosophc- neutrosophc complex lke vector spce, 19- O Ordnry complex neutrosophc lner lgebr, 6-8 Ordnry complex neutrosophc subspce, 3- Ordnry complex neutrosophc vector spce, 17-8 I Pseudo complex modulo nteger lner sublgebr, Pseudo complex strong neutrosophc complex modulo nteger vector subspce / lner sublgebr, 16-8 Pseudo double lner sublgebrs, 69-7 Pseudo specl neutrosophc subsemgroup complex neutrosophc lner sublgebr, 6-5 Pure neutrosophc nteger, complex ntegers,

220 I Rtonl neutrosophc complex number, 7-9 Rel neutrosophc complex number, 7-9 I S-complex neutrosophc modulo nteger lner lgebr, Semgroup complex neutrosophc double lner lgebr, 67-9 Semgroup complex neutrosophc lner lgebr, 9-51 Semgroup complex neutrosophc lner lgebr, 66-7 Semgroup complex neutrosophc modulo nteger vector spce, Semgroup fuzzy neutrosophc complex semvector spce, 81- Semgroup neutrosophc complex double lner sublgebr, 68-7 Semgroup neutrosophc complex vector spce, - Set complex neutrosophc lner lgebr, 5-6 Set complex neutrosophc vector spce, 33- Set neutrosophc complex modulo nteger lner lgebr, Set neutrosophc complex modulo nteger vector spce, 17-8 Smrndche complex modulo nteger vector spce, 113- Smrndche complex neutrosophc modulo nteger vector spce, Specl complex neutrosophc lke vector spce, 1-3 Specl neutrosophc complex vector spce, -5 Specl semgroup neutrosophc complex vector spce, 3-5 Specl set vector spce of complex neutrosophc rtonls, 38-9 Subrng of fnte complex modulo ntegers, 9-19

221 ABOUT THE AUTHORS Dr.W.B.Vsnth Kndsmy s n Assocte Professor n the Deprtment of Mthemtcs, Indn Insttute of Technology Mdrs, Chenn. In the pst decde she hs guded 13 Ph.D. scholrs n the dfferent felds of non-ssoctve lgebrs, lgebrc codng theory, trnsportton theory, fuzzy groups, nd pplctons of fuzzy theory of the problems fced n chemcl ndustres nd cement ndustres. She hs to her credt 66 reserch ppers. She hs guded over 68 M.Sc. nd M.Tech. projects. She hs worked n collborton projects wth the Indn Spce Reserch Orgnzton nd wth the Tml Ndu Stte AIDS Control Socety. She s presently workng on reserch project funded by the Bord of Reserch n Nucler Scences, Government of Ind. Ths s her 58 th book. On Ind's 6th Independence Dy, Dr.Vsnth ws conferred the Klpn Chwl Awrd for Courge nd Drng Enterprse by the Stte Government of Tml Ndu n recognton of her sustned fght for socl justce n the Indn Insttute of Technology (IIT) Mdrs nd for her contrbuton to mthemtcs. The wrd, nsttuted n the memory of Indn-Amercn stronut Klpn Chwl who ded bord Spce Shuttle Columb, crred csh prze of fve lkh rupees (the hghest prze-money for ny Indn wrd) nd gold medl. She cn be contcted t vsnthkndsmy@gml.com Web Ste: or Dr. lorentn Smrndche s Professor of Mthemtcs t the Unversty of New Mexco n USA. He publshed over 75 books nd rtcles nd notes n mthemtcs, physcs, phlosophy, psychology, rebus, lterture. In mthemtcs hs reserch s n number theory, non- Euclden geometry, synthetc geometry, lgebrc structures, sttstcs, neutrosophc logc nd set (generlztons of fuzzy logc nd set respectvely), neutrosophc probblty (generlzton of clsscl nd mprecse probblty). Also, smll contrbutons to nucler nd prtcle physcs, nformton fuson, neutrosophy ( generlzton of dlectcs), lw of senstons nd stmul, etc. He cn be contcted t smrnd@unm.edu

222