Sequences of Intuitionistic Fuzzy Soft G-Modules

Transcription

1 Interntonl Mthemtcl Forum, Vol 13, 2018, no 12, HIKARI Ltd, wwwm-hkrcom Sequences of Intutonstc Fuzzy Soft G-Modules Velyev Kemle nd Huseynov Afq Bku Stte Unversty, Bku, Azerjn Copyrght 2018 Velyev Kemle nd Huseynov Afq Ths rtcle s dstruted under the Cretve Commons Attruton Lcense, whch permts unrestrcted use, dstruton, nd reproducton n ny medum, provded the orgnl work s properly cted Astrct The noton of ntutonstc fuzzy G -modules nd the noton of exct sequence of ntutonstc fuzzy G -modules ws ntroduced y PK Shrm [13, 15] nd studes ther propertes In ths pper we develop the noton of exct sequence of ntutonstc fuzzy soft G -modules nd studes ther propertes Keywords: Intutonstc fuzzy set, ntutonstc fuzzy soft set, ntutonstc fuzzy G -sumodule, ntutonstc fuzzy soft G -sumodule, exct sequence 1 Introducton Mny prctcl prolems n economcs, engneerng, envronment, socl scence, medcl scence etc cnnot e delt wth y clsscl methods, ecuse clsscl methods hve nherent dffcultes The reson for these dffcultes my e due to the ndequcy of the theores of prmeterzton tools Molodtsov [12] ntted the concept of soft set theory s new mthemtcl tool for delng wth uncertntes Mj et l [11] presented the concept of fuzzy soft set The theory of fuzzy sets, frst developed y Zdeh n [18], s perhps the most pproprte frmework for delng wth uncertntes, numer of generlztons of ths fundmentl concept hve come up The noton of ntutonstc fuzzy sets ntroduced y Atnssov [3] s one mong them Algerc structures ply vtl role n Mthemtcs nd numerous pplctons of these structure re seen n mny dscplnes such s computer scences, nformton scences, theoretcl physcs, control engneerng nd so on Ths nspres reserchers to study nd crry out reserch n vrous concepts of strct lger n fuzzy settng Bsws [4] ppled the concept of ntutonstc fuzzy sets to the theory of groups nd studed ntutonstc fuzzy sugroup of group Fuzzy sumodules of module M over rng R were frst ntroduced [10] Aktş nd Çğmn [2] defned soft groups nd compred soft sets wth fuzzy sets nd rough sets F Feng et l [5] gve soft semrngs nd UAcr et l [1] ntroduced ntl concepts of soft rngs

2 538 Velyev Kemle nd Huseynov Afq After tht the defnton of fuzzy soft group ws gven y some uthors [8] Qu- Me Sun et l [16] defned soft modules nd nvestgted ther sc propertes Gunduz (Ars) nd ByrmovS n [6],[7] dd some nvestgtons on ntutonstc fuzzy soft modules The noton of ntutonstc fuzzy G -modules ws ntroduced y PK Shrm [13] KM Velyev nd SA Byrmov ntroduced fuzzy soft G-modules [17] Mny propertes lke representton, reduclty, complete reduclty nd njectvty of ntutonstc fuzzy G -modules hve een dscussed n [13],[14],[15] In ths pper we develop the noton of exct sequence of ntutonstc fuzzy soft G -modules nd studes ther propertes 2 Prelmnres Defnton 21 ([12]) Let X e n ntl unverse set nd E e set of prmeters A pr F, E s clled soft set over X f nd only f F s F : E P X, mppng from E nto the set of ll susets of the set X, e, where P X s the power set of X Defnton 22 ([16]) Let F, A e soft set over M A F for ll x A F, s sd to e soft module over M f nd only f x M Defnton 23 ([7]) Let IFS X denote the set of ll ntutonstc fuzzy sets on X nd A E A pr F, A s clled n ntutonstc fuzzy soft set over X, where F s mppng from A nto IFS X Tht s, for ech A, F F, F : X I s n ntutonstc fuzzy set on X, where F, F : X I re fuzzy sets Defnton 24 ([7) Let F, A e n ntutonstc fuzzy soft set over M Then F, A s sd to e n ntutonstc fuzzy soft module over M ff A, F, F F s n ntutonstc fuzzy sumodule of M Defnton 25 ([7]) Let F, A nd H, B e two ntutonstc fuzzy soft modules over M nd N respectvely, nd let f : M N e homomorphsm of modules, nd let g : A B e mppng of sets Then we sy tht f, g: F, A H, B s n ntutonstc fuzzy soft homomorphsm of ntutonstc fuzzy soft modules, f the followng condtons re stsfed: g f F H, f g F H We sy tht F, A s n ntutonstc fuzzy soft homomorphc to H, B g Note tht for A, f : M, F, F N, H, H s n ntutonstc g fuzzy homomorphsm of ntutonstc fuzzy modules

3 Sequences of ntutonstc fuzzy soft G - modules 539 Defnton 26 ([13]) Let G e group nd M e G -module over K, whch s sufeld of C Then ntutonstc fuzzy G -module on M s n ntutonstc A, of M such tht followng condtons re stsfed fuzzy set A A () x y x y nd,, A A A nd x, y M x y x y K A A A () A gm Am nd A A, ; Defnton 27 (15]) For ny IFS A A gm m g G m M A x, x, x : x X of set X We denote the support of the IFS set A y A nd s defned s A x X : x 0 nd x 1 A Proposton 21 ([15]) Let f : X Y e mppng nd AB, re IFS of X nd Y respectvely Then the followng result holds f A f A nd equlty hold when the mp f s jectve () 1 1 () f B f B 3 Sequences of ntutonstc fuzzy soft G-modules Let F, A e ntutonstc fuzzy soft G -module on M, GB, e ntutonstc fuzzy soft G -module on N f : M N s G -modules homomorphsm, : A Bs mppng of sets Defnton 31 If for ech A f : M, F, F s homomorphsm of the ntutonstc fuzzy G -modules, then f, pr s clled homomorphsm of ntutonstc fuzzy soft G -modules f, : F, A G, B Let f, : F, A G, B A e homomorphsm of IFSG(M) nd ker f M e the kernel of M, such s G sumodule Defne structure of IFSG(M) on ker f followng F, F F, F F Kerf, F for A In ths wy show IFSG(M) structure of Im f G, G G, G G Im f, G F Theorem 31 Let M nd N e G -modules nd let Kerf N,such s G sumodule of N: G Kerf f e G -module homomorphsm If GB, s ntutonstc fuzzy G -module on N, 1 then f G, B s n ntutonstc fuzzy soft G-module over M 1 Proof If the mppng f G : B IFSG( M ) s defned y

4 540 Velyev Kemle nd Huseynov Afq 1, f G x G f x 1 f G x G f x For k1, k2 K, x, ym nd g G, we hve 1 f G k x k y G f k x k y G k f x k f y G f x G f y f G x f G y Thus, 1 1 f G k x k y f G x f 1 G y 1 2 Smlrly, we cn show tht, f G k x k y G f k x k y G k f x k f y G f x G f y f G x f G y Thus, f G k x k y f G x f G y Also, Thus, f 1 G gm f 1 G m 1 2 f 1 G gm G f gm G g f m G f m f 1 G m 1 1 Smlrly, we cn show tht, f G gm f G m 1 Hence f G, B s ntutonstc fuzzy G -module on M Theorem 32 Let M nd N e G -modules nd f : M N e G -module homomorphsm If F, A s n ntutonstc fuzzy G -module on M, then f F, A s n ntutonstc fuzzy soft G -module on N Proof If the mppng f : F : A IFSG( N) s defned y f F y sup F x: f x y, nf : Now we show tht, For k1, k2 K,, f F y F x f x y f F A s n ntutonstc fuzzy soft G -module on N x y N nd g G, we hve 1 f F k1x k2 y F z : z f k1x k2 y F z : f z k x k y, x, y N, z M :, :, F k z k z f z x f z y F k z k z z f x z f y 1 1 F z F z : z f x, z f y 1 1 F z : z f x F z : z f y f F x f F y Thus, f F k x k y f F x f F y 1 2

5 Sequences of ntutonstc fuzzy soft G - modules 541 Smlrly, we cn show tht, Also, f F k x k y f F x f F y F gm : gm M, f gm gn, g G, n N, m M 1 f F gn f F n f F gn f F n, f F gn F x : x f gn F x : f x gn, g G, nn F m : m M, f m n N F m : m f n f F n Thus, Smlrly, we cn show tht Hence Let, f F A s n ntutonstc fuzzy soft G -module on N F A e fmly of soft G -module over M I nd let the prmeters set I e fxed pont We denote the fxed pont of A s 0 F 0 0 For A I A nd M M we defne the mppng F : A M y F F, for ll I nd let A Then, Defnton 32, I F, A Theorem 33 If, I over M, then, I I F A M I Proof Defne F : I A I M for ll F j F, j F where : F A s soft G -module over M F A s sd to e drect sum of F, A nd denoted s I F A s fmly of ntutonstc fuzzy soft G -modules mppng Snce, M for ll I, I I s n ntutonstc fuzzy soft G -modules over A y I j M M s n emeddng I j F j F -s n ntutonstc fuzzy G -module over F s n ntutonstc fuzzy G -sumodule over I M Defnton 33 For ny IFSS F, A of set X We denote the support of the IFSS y F, A nd s defned s F x X, F x 0, F x 1 It s cler tht F, A s soft set on X Proposton 31 Let f : X Y, : A B e two mppngs nd F, A,, G B re IFSS of X nd Y respectvely Then the followng result holds: ) f, F, A f F, A

6 542 Velyev Kemle nd Huseynov Afq 1 1 ) f G, B f G, A Theorem 34 ) Let F, A s IFSG(M) Then F, A ) For ny F, A,, F, A G, B F, A G, B s soft G -sumodule of M G B soft G -module of M, we hve F, A G, B F, A G, B c) Defnton 34 We defne two IFSS, A nd M, Aof M s For And x M 1,0, x 0 x ; M x 1,0 0,1, x 0 Then the IFSS, A M, A re IFSSM, of M whch re ctully equvlent of, 0 nd M n module theory Defnton 35 If F, A, G, B IFSGM clled the drect sum of F, A nd, on M, then the sum F, A G, B GB f F, A G, B, A B wrte t s s F, A G, B Theorem 35 Let F, A, G, B, H, C F, A G, B H, C F, A G, B H, C Let, then s nd we soft G -modules of M such tht f 1 f f 1 f 2 M 1 M M 1 (1) re sequence of G -modules nd G -module homomorphsm Defnton 36 Let M, Z e G -modules nd let F, AIFSG M, Z Suppose tht (1) s exct sequence of G -modules Then the sequence f 1,1 A f,1 1,1 2,1 1, A f, A f F 1, A A F A F A (2) of ntutonstc fuzzy soft G -module s sd to e exct t, for ll A M, F M, F M, F the sequence of ntutonstc fuzzy G -module s exct F, A, G, B IFSG M Theorem 36 Let e such tht F, A G, B drect sum of ntutonstc fuzzy soft sumodules of G -module M so tht F, A G, A s drect sum of soft G -modules Then the sequence,1 A,1A F, A IFSG F nd G, A IFSG F 0 F, A F, A G, A G, A 0 s exct, consderng Proof Note tht the sequence of soft G -modules s n exct sequence s

7 Sequences of ntutonstc fuzzy soft G - modules 543,1 A,1A F A F A G A G A 0,,,, 0,1 A nd,1 A re respectvely the cnoncl njecton nd projecton We hve to prove tht the sequence,1 A,1A 0 F, A F, A G, A G, A 0 s n exct sequence, where of ntutonstc fuzzy soft G -modules For ech A let F G F x Then F x F x, F x, where 1 F :,,, t t F t x f x F x f x F otherwse 0, f x F 0, nd F x f x F F x 1, 1 F t : t F, t x, f x, otherwse 1, f x F Thus, F, A F, A x F Also, for A F Gx F G x, F G x (1), where F y G z : y, z M, y z x, f x y z F G x 0, f x y z nd F x, f x F 0, f x F F y G z : y, z M, y z x, f x y z F G x 1, f x y z F x, f x F 1, f x F [Note tht F, A G, A s drect sum, so,, wth x F, then the only posslty s xx 0 or x y z; y, z F But n the second cse G z 0, G z 1] Thus, F, A G, A F, A f x F for A It follows from (1) nd (2) tht F, A F, A G, A For xg, F Gx F G x, F G x where, F A G A If x y z (2)

8 544 Velyev Kemle nd Huseynov Afq F G x F G t : t F G ; t x F G r x : r F [Snce : F G G projecton] F r G x : r F G x [Snce 1 Smlrly, we hve F G x G x Hence F G G Now y (1), we hve F x, F x, f xf ker F, Ax e, 0,1, f xf ker F, A ker Therefore, F A F A G A G A -s the F r wth r 0 ] 0,,,, 0 s n exct sequence of ntutonstc fuzzy soft G -modules f g Theorem 37 Let M N P e sequence of G -modules exct t N nd let F, A IFSG M, G, A IFSG N, H, A IFSG P Then the f,1 A g,1a sequence F, A G, A H, A of ntutonstc fuzzy soft G - modules s exct t GAonly, f for ech A the sequence f g F G H s sequence of G -modules exct t G, where f nd g re restrcton of f nd g to F nd G Proof Suppose tht, f g F A G, A H, A s exct t, defnton f F, A G, A, g G, A H, A nd f F A Now, consder the sequence f g F G H We clm tht ths sequence s exct t G f F x 0 x f F, A, A: For 1 respectvely GA Then y, ker g nd f F x F t : f t x, t M 0 nd F t : f t x, t M 1 st 1, t2m such tht x f t1 f t2, F t1 0, F t2 1 (As F t1 F t2 1 lwys, so f F t1 0 then st 1 M such tht x f t1, F t1 0 nd F t1 1 e, t1 F x f t1 f F Thus, we get f F, A f F Smlrly, we get g F, A g F Therefore, f F f F f F, A G, A s f F, A G, A F t ) 2 1

9 Sequences of ntutonstc fuzzy soft G - modules 545 Smlrly, g G g G, A H, A Now, Snce f F, A ker g t follows tht f F ker g Thus, the sequence f g F G H s exct t G Ths completes the proof of the theorem 4 Conclusons The mn focus of ths rtcle s to ntroduce the concept of exct sequence of G - modules y ntutonstc soft fuzzfcton the concept n crsp theory We develop the noton of exct sequence of ntutonstc fuzzy soft G -modules nd studes ther propertes References [1] U Acr, F Koyuncu, B Tny, Soft sets nd soft rngs, Comput Mth Appl, 59 (2010), [2] H Aktş, N Çğmn, Soft sets nd soft group, Informton Scence, 177 (2007), [3] KT Atnssov, Intutonstc fuzzy sets, Fuzzy Sets nd Systems, 20 (1986), no 1, [4] R Bsws, Intutonstc fuzzy sugroups, Mthemtcl Forum, 10 (1989), [5] F Feng, YB Jun, X Zho, Soft semrngs, Comput Mth Appl, 56 (2008), [6] C Gunduz, S Byrmov, Fuzzy soft modules, Interntonl Mthemtcl Forum, 6 (2011), no 11, [7] C Gunduz, S Byrmov, Intutonstc fuzzy soft modules, Comput Mth Appl, 62 (2011), [8] L Jn-lng, Y Ru-x, Y Bng-xue, Fuzzy soft sets nf fuzzy soft groups, Chnese Control nd Decson Conference, (2008), [9] SR Lonpez-Permouth nd DS Mlk, On ctegores of fuzzy modules, Inform Sc, 52 (1990),

10 546 Velyev Kemle nd Huseynov Afq [10] PK Mj, R Bsms, AR Roy, Soft set theory, Comput Mth Appl, 45 (2003), [11] PK Mj, R Bsms, AR Roy, Fuzzy soft sets, The Journl of Fuzzy Mthemtcs, 9 (2001), no 3, [12] D Molodtsov, Soft set theory- frst results, Comput Mth Appl, 37 (1999), [13] PK Shrm nd Trndeep Kur, İntutonstc fuzzy G-modules, Notes on İntutonstc Fuzzy Sets, 21 (2015), no 1, 6-23 [14] PK Shrm, Drect Sum of Intutonstc Fuzzy Sumodules of G-module, Asn Journl of Fuzzy nd Appled Mthemtcs, 4 (2016), no 4, [15] PK Shrm, Exct Sequence of İntutonstc fuzzy G-modules, Notes on İntutonstc Fuzzy Sets, 23 (2017), no 5, [16] Qu-Me Sun, Z-Long Zhng, Jng Lu, Soft sets nd soft modules, n Interntonl Conference on Rough Sets nd Knowledge Technology, Vol 5009, Sprnger, 2008, [17] KM Velyev, SA Byrmov, Fuzzy Soft G-modules, News of Bku Unversty, Physco-Mthemtcl Scences, 2018, [18] LA Zdeh, Fuzzy sets, Informton nd Control, 8 (1965), Receved: Octoer 23, 2018; Pulshed: Decemer 3, 2018