Fuzzy soft -ring. E Blok Esenler, Istanbul, Turkey 2 Department of Mathematics, Marmara University, Istanbul, Turkey

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1 IJST (202) A4: Irnn Journl of Scence & Technology Fuzzy soft -rng S Onr, B A Ersoy * nd U Tekr 2 Deprtment of Mthemtcs, Yıldız Techncl Unversty Dvutpş Kmpüsü E Blok Esenler, Istnbul, Turkey 2 Deprtment of Mthemtcs, Mrmr Unversty, Istnbul, Turkey E-mls: serkn0r@gmlcom, ersoy@yldzedutr & utekr@mrmredutr Abstrct The concept of fuzzy soft Γ-rng s ntroduced; nd some propertes of fuzzy soft Γ-rngs re gven Then the defntons of fuzzy soft Γ-dels re proposed nd some of ther theores re consdered Keywords: rng; fuzzy soft rng; soft rng Introducton Snce the concept of soft sets ws ntroduced by Molodtsov [] n 999, soft sets theory hs been extensvely studed by mny uthors Ths theory hs been ppled to mny dfferent felds, such s functon smoothness, emnn nd Perron ntegrton, mesurement nd gme theory, decson mkng M et l [2, 3] ponted out severl drectons or pplctons of soft sets They lso studed severl opertons on the theory of soft sets Chen et l [4] ntroduced new defnton of soft set prmeterztons reducton, nd compred ths defnton to the relted concept of ttrbutes reducton n rough set theory Akts et l[5] studed the bsc concept of soft set theory, nd compred soft sets to fuzzy nd rough sets, provdng some exmples to clrfy ther dfferences The lgebrc structure of set theores delng wth uncertntes hs been studed by some uthors Akts et l [5] ppled the noton of set to the theory groups Jun [6] ntroduced the notons of soft BCK/BCI-lgebrs, nd then nvestgted ther bsc propertes [7] Öztürk et l [8] dscussed new vew of fuzzy Gmm rngs It s well known tht the concept of fuzzy sets, ntroduced by Zdeh [9], hs been extensvely ppled to mny scentfc felds In 97, osenfeld [0] ppled the concept to the theory of groupods nd groups In 982, Lu [] defned nd studed fuzzy subrngs s well s fuzzy dels Snce then mny ppers concernng vrous fuzzy lgebrc structures hve ppered n the lterture The vrous constructons of fuzzy quotents rngs nd fuzzy somorphsm hve been nvestgted respectvely by severl reserchers (see eg [2, 3, nd 4]) Also, M et l presented the defnton of fuzzy soft set, nd oy et l presented some pplctons of ths noton to the decson- mkng problems n [2] Inn et l hve lredy ntroduced the defnton of fuzzy soft rngs nd studed some of ther bsc propertes In ths pper, we ttempt to study fuzzy soft rng theory by usng fuzzy soft sets We frst ntroduce fuzzy soft rngs generted by fuzzy soft sets, nd gve ther propertes Consequently we study the defnton of fuzzy soft del nd derve some results from them, respectvely 2 Prelmnres In ths secton, for the ske of completeness, we frst cte some useful defntons nd results Defnton 2 [9] A fuzzy subset n set X s : X 0, functon Defnton 22 [] Let U be n ntl unverse E set of prmeters nd A E Let PU denote the fuzzy power set of U A pr F, A s clled fuzzy soft set over U, where F s mppng F: A P U gven by *Correspondng uthor eceved: 3 Februry 202 / Accepted: 0 Mrch 202

2 IJST (202) A4: F, A nd GB, be two fuzzy soft sets over U Then F, A s sd to be fuzzy soft subset of GB, f A B Fx s fuzzy subset of x A Defnton 23 [] Let Gx for ll We denote the bove ncluson reltonshp by F, A G, B Smlrly,, fuzzy soft superset of GB, f, soft subset of, be denoted by F, A G, B F A s clled GB s fuzzy F A The bove reltonshp cn Defnton 24 [5] The ntersecton of two fuzzy soft sets F, A nd GB, over U s the soft set H, C, where C A B nd for ll x C, ether H x F x or H x G x Ths ntersecton s denoted by ( F, A) ( G, B) ( H, C) Defnton 25 [5] The unon of two fuzzy soft sets, GB, over U s the soft set F A nd H, C, where C A B nd for ll x C, defne by: F( x) f x A -B H ( x) G( x) f x B -A F( x) G( x) Otherwse The bove reltonshp s denoted by F, AG, BH, C Defnton 26 [5] Let F, A nd GB, be two fuzzy soft sets Then we denote F, A AND G, B by F, A G, B The soft set F, A G, B s defned by ( H, A B), where H(, ) F( ) G( ), for ll (, ) A B Defnton 27 [5] Let F, A nd GB, be two fuzzy soft sets Then F, A O G, B denoted by F, A G, B s defned by ( H, A B), where H(, ) F( ) G( ), for ll (, ) A B Now, we show the defnton nd del of rng nd of rng Defnton 28 [5] Let S nd be two ddtve beln groups S s clled rng f there exst mppng SS S by,, b b stsfyng the followng condtons: ( b) cc bc, 2 ( b c) b c, 3 ( ) bb b, 4 ( bc) ( b) c for ll bc,, S nd for ll, A left (resp rght) del of rng S s subset A of S whch s n ddtve subgroup of S nd SA A (resp, SA A) where, SAxy x, ys, If A s both left nd rght del, then A s clled gmm del of S Defnton 28 [6] Let S nd K be two rngs, nd f be mppng of S nto K Then f s clled homomorphsm f f b f f b f b nd for ll b, S nd f f b Defnton 28 [5] A fuzzy set n rng S s clled fuzzy del of S, f for ll x y S, the followng requrements re stsfed: x y mn x, y 2 xy mx x, y 3 Fuzzy soft rng nd In wht follows let A be rng nd nonempty set nd wll refer to n rbtrry ternry relton mong one element of A, n element of nd n element of A, tht s, s subset of A A, unless otherwse specfed A set vlued functon f : N P A cn be defned f { baû b ; } s for ll

3 47 IJST (202) A4: -2 N The pr f, N s then soft set over A Defnton 3 Let f, N be soft set over rng A Then f, N s clled soft rng over A ff f s sub rng of A, for ll N A Defnton 32 Let,, be rng nd E be prmeter set nd A E Let f be mppng gven by f : AA 0,, where 0, denotes the collecton of ll fuzzy subsets of, f : AA 0,,, b f( b) f b f f : 0, { b} b s clled fuzzy soft rng over f nd only f for ech b, And the correspondng fuzzy subset f of f s fuzzy soft subrng of, e x, y f( x f( x) f( f( x) f( x) f ( x f ( x) f ( Theorem 33 Let ( f, N ) be fuzzy soft set over, then ( f, N ) be fuzzy soft rng over ff for ech N nd x, y the followng condtons hold f( x f( x) f( f ( x f ( x) f ( Proof: Frst we suppose tht ( f, N ) be fuzzy soft rng over Then for ech N nd x, y f( x f( x( ) f( x) f( f ( x) f ( conversely, suppose tht the gven condtons hold Then for ech N, x, y f(0) f( x x) f( x) f( x) f ( x) where 0 s the ddtve dentty of f( x) f(0 x) f(0) f( x) f( x) f( x) f ( x) Now, f( x f( x( ) f( x) f( f( x) f( f ( x f ( x) f ( so ths completes proof Exmple 34 Z 6 0,, 2, 3, 4, 5 defne x y x y for ll x, y, Then s rng nd let N A A f : N P( ) be set vlued functon defned 0,08,,02, 2,0,4, by f 3,02,4,0,4,5,02 0,0,9,,05, 2,05, f b 3,0,, 4,05, 5,05 nd 0,0,7,,03, 2,03, f c 3,0,3, 4,03, 5,03 Obvously ( f, N ) s fuzzy soft set over Also, we see tht f s fuzzy del of for ll N, thus ( f, N ) s fuzzy soft rng over Defnton 35 Let ( f, N ) nd ( g, M ) be two fuzzy soft rngs over "( f, N) AND( g, M )", denoted by ( f, N) ( g, M), s defned by ( f, N) ( g, M) ( h, S) where S N M, hs fs gs for ll s S

4 IJST (202) A4: Theorem 36 If ( f, N ) nd ( gm, ) be two fuzzy soft rngs over, then ( f, N) ( g, M) s fuzzy soft rng over where Proof: ( f, N) ( g, M) ( h, S) S N M, for ll ss, hs fs gs x, y nd, hs( x fs( x gs( x ( fs( x) fs( ) ( gs( x) gs( ) ( fs( x) gs( x)) ( fs( gs( ) ( fs gs)( x) ( fs gs)( hs( x) hs( hs( x hs( x) hs( hs( x fs( x gs( x fs( x) fs( ) ( gs( x) gs( ) ( fs( x) gs( x)) ( fs( gs( ) ( fs gs)( x) ( fs gs)( hs( x) hs( h ( x h ( x) h ( s s s Thus h s s fuzzy soft sub rng of for ll ss N M nd so ( f, N) ( g, M) s fuzzy soft rng over Defnton 37 Let ( f, N) nd ( g, M ) be two fuzzy soft rngs over "( f, NOgM ) (, )", denoted by ( f, N) ( g, M), s defned by ( f, N) ( g, M) ( h, S) where S N M, hs fs gs for ll s S Theorem 38 If ( f, N ) nd ( gm, ) be two fuzzy soft rngs over, Then ( f, N) ( g, M) s fuzzy soft rng over Proof: It cn be smlr proof of Theorem 36 nd for ll S, ether f h or g h Ths ntersecton s denoted by ( f, N) ( g, M) ( h, S) Theorem 30 Let ( f, N ) nd ( gm, ) be two fuzzy soft rngs over Then ( f, N) ( g, M) s fuzzy soft rng over Proof: Let ( f, N) ( g, M) ( h, S), where S N M, S, then xy,, h( x f( x g( x f( x) f( g( x) g( ( f( x) g( x)) ( f( g( ) ( f g)( x) ( f g)( h( x) h( where f, g, h re the fuzzy soft subsets of correspondng to the prmeter S h( x f( x g( x f( x) f( g( x) g( ( f( x) g( x)) ( f( g( ) ( f g)( x) ( f g)( h ( x) h ( ths completes the proof Defnton 3 The unon of two fuzzy soft rngs ( f, N ) nd ( g, M ) over be denoted by ( f, N) ( g, M), we defne ( f, N) ( g, M) ( h, C), where C N M nd c C fc, f c N - M hc gc, f c M - N f g, f cn M for ll, c c x y nd Theorem 32 Let ( f, N ) nd ( gm, ) be two fuzzy soft rngs over Then ( f, N) ( g, M) s fuzzy soft rng over, f N M Defnton 39 The ntersecton of two fuzzy soft rngs ( f, N ) nd ( gm, ) over s the fuzzy soft rng ( hs, ), where S N M,

5 473 IJST (202) A4: -2 Proof: We know tht ( f, N) ( g, M) ( h, C), where C N M nd cc fc, f c N - M hc gc, f c M - N f g, f cn M c c for ll x, y nd For cn M, hc( x fc( x fc( x) fc( Snce cn M, we sy tht hc( x) fc( x) nd hc( fc( hc( x hc( x) hc( hc( x fc( x fc( x) fc( hc( x) hc( hc( x hc( x) hc( for cm N hc( x gc( x gc( x) gc( hc( x) hc( hc( x hc( x) hc( hc( x gc( x gc( x) gc( hc( x) hc( h ( x h ( x) h ( c c c Defnton 33 Let ( f, N) nd ( g, M ) be two fuzzy soft rngs over The fuzzy soft rng ( g, M ) s clled fuzzy soft sub rng of ( f, N ), f t stsfes: M N g s sub rng of f, for ll M Defnton 34 Let ( f, N) I be fuzzy soft rngs over Then the ntersecton of these fuzzy soft rngs s defned s beng the fuzzy soft rng ( g, M ) stsfyng the followng condtons: M A I 2 For ll M, there exst n 0 I such tht g f ( ) In ths cse, we wrte 0 ( f, N ) ( g, M) I Defnton 35 Let ( f, N) ı be fuzzy soft rngs over Then, ( gm, ) ( f, N) s fuzzy soft rng such tht I M N nd g f, I I I M for ll ( ) 2 ( gm, ) ( f, N) s fuzzy soft rng such tht I M N nd g f, I I I M ll ( ) for Theorem 36 Let ( f, N ) be fuzzy soft rng over If ( f, N) I s nonempty fmly of fuzzy soft sub rng of ( f, N ), where I s n ndex set, then, s fuzzy soft sub rng of ( f, N ) I ( f, N ) 2 ( f, N ) I s fuzzy soft sub rng of ( f, N ) 3 ( f, N ) I s fuzzy soft sub rng of ( f, N ), where N N for ll, I Proof: Usng Defnton 34 nd snce ( f, N ) s fuzzy soft sub rng of ( f, N ) for ll I, we hve g : M P( ) by g f, for ll M N N nd I In ths cse,, I f s sub rng of for ll M nd I, so f, s sub rng of Thus I ( g, M ) s fuzzy soft rng over Hence ( gm, ) ( f, N) s fuzzy soft sub rng I of ( f, N ) by Defnton 34 The proofs of 2 nd 3 cn be wrtten smlrly

6 IJST (202) A4: Defnton 37 Let ( f, N ) be fuzzy soft rng over Then ( f, N ) be fuzzy soft del over ff for ech N nd xy,, the followng condtons hold: f ( x f ( x) f ( f( x f( x) f( defned by ( f, N) Theorem 38 For ny fuzzy soft dels ( f, N) nd ( g, M ) over, where N M, we hve ( f, N) ( g, M) Proof: Usng Defnton 39 we cn wrte ( f, N) ( g, M) ( h, S), where S N M nd for ll ss NM, hs fs gs x, y, hs( x fs( x gs( x ( fs( x) fs( ) ( gs( x) gs( ) ( fs( x) gs( x)) ( fs( gs( ) snce ( f, N) nd ( gm, ), we know tht ll f s nd g s re fuzzy soft del of for s S Therefore, hs( x hs( x) hs( And, hs( x fs( x gs( x ( fs( x) fs( ) ( gs( x) gs( ) ( fs( x) ( gs( x)) ( fs( gs( ) fs gs( x) fs gs( hs( x) hs( Hence ( hs, ) s fuzzy soft del nd therefore ( f, N) ( g, M) ( h, S) Theorem 39 For ny fuzzy soft dels ( f, I ) nd ( gj, ) over, n whch I nd J re dsont, we hve ( f, I) ( g, J) nd Proof: Assume tht ( f, I) ( gj, ) ( f, I) ( g, J) ( h, K) where K I J By mens of defnton we cn wrte nd for every K f, f c I - J h() c g, f cj - I f g, f ci J snce I J s ether ci J or c J I for ll c K If ci J, then h f s the del of snce ( f, I) If c J I, then h g s the del of snce ( gj, ) Thus h s the del of for ll c K, nd so ( f, I) ( g, J) ( h, K) Theorem 320 Let ( f, N ) be fuzzy soft del over If ( f, I ) J s nonempty fmly of fuzzy soft dels of ( f, N ), where J s n ndex set, then, ( f, I ) J 2 ( f, I ) J 3 ( f, I ) J where I I 0 ll, k J k for Defnton 32 Let ( f, N) nd ( g, M ) be two fuzzy soft rngs over A nd B, respectvely Let F : A B nd G: N M be two functons Then the pr ( FG, ) s clled fuzzy soft rng homomorphsm f t stsfes the followng condtons: F s n onto rng homomorphsm G s n onto rng homomorphsm F( f) ( g) G for ll N If there exst fuzzy soft rng homomorphsm between ( f, N) nd ( gm, ), we sy tht ( f, N ) s fuzzy soft homomorphc to ( gm, ), nd s denoted by ( f, N)~ ( g, M) Moreover, F s n somorphsm nd ( f, N ) s fuzzy soft homomorphc to ( g, M ), whch s denoted by ( f, N) ( g, M) Now, we show tht the homomorphc mge nd premge of fuzzy soft rng re lso fuzzy soft rng

7 475 IJST (202) A4: -2 Defnton 322 Let : X Y nd : N M be two functons, where N nd M re prmeter sets for fuzzy soft sets X nd Y, respectvely Then the pr (, ) s clled fuzzy soft functon from X to Y Defnton 323 Let ( f, N) nd ( g, M ) be two fuzzy soft rngs over X nd Y Let (, ) be fuzzy soft functon from X to Y The mge of ( f, N ) under the fuzzy soft functon (, ) denoted by (, )( f, N) s the fuzzy soft rng over Y defned by (, )( f, N) ( ( f), ( N)) where, - f( x), f x ( ( x) y( ) k ( f) k ( 0, otherwse k( N), y Y 2 The premge of ( g, M ) under the fuzzy soft functon (, ) (, ) ( gm, ) denoted by s the fuzzy soft rng over X defned by (, ) ( gm, ) ( ( g), ( M)) where, x X ( g) ( x) g ( ) ( ( x)) ( M) If nd re nectve (surectve), then (, ) s sd to be nectve (surectve) Defnton 324 Let (, ) be fuzzy soft functon from X to Y If s homomorphsm functon from X to Y, then (, ) s sd to be fuzzy soft homomorphsm If s somorphsm functon from X to Y nd s one to one mppng from N onto M, then (, ) s sd to be fuzzy soft somorphsm Theorem 325 Let ( f, N ) be fuzzy soft rng over nd (, ) be fuzzy soft homomorphsm from to S Then (, )( f, N) s fuzzy soft rng over S Proof: Let k ( N) ( y ) or ( y ) nd y, y2 2 Y If the proof s cler ( x ) y, ( x2) y2 Let us ssume tht ( f) ( y y ) f ( k) k 2 ( xx2) yy2 ( ) k f ( x x ) ( ) k ( ) k 2 ( f ( x ) f ( x )) 2 f ( x ) f ( x ) 2 ( ) k ( ) k ths nequlty s stsfed for ech x, x 2 X, where ( x) y, s stsfed ( x2) y2 Then we hve: ( f) ( y y ) k 2 f( x) f( x2) ( x ) y ( ) k ( x ) y ( ) k 2 2 ( f) ( y ) ( f) ( y ) k k 2 nd smlrly we hve ( f ) ( yy ) ( f) ( y ) ( f) ( y ) k 2 k k 2 Theorem 326 Let ( g, M ) be fuzzy soft rng over S nd (, ) be fuzzy soft homomorphsm from to S Then (, ) ( gm, ) s fuzzy soft rng over Proof: Let ( B) nd x, x2 X, ( g) ( xx2 ) g ( ) ( ( xx2)) g ( ) ( ( x) ( x2)) g ( ( x ) g ( ( x ) ( ) ( ) 2 g x g x2 ( ) ( ) ( ) ( ) nd smlrly we hve ( g) ( x x ) ( g) ( x ) ( g) ( x ) 4 Concluson 2 2 In ths work the theoretcl pont of vew of fuzzy sets n rng nd del re dscussed The work s focused on fuzzy soft rngs nd fuzzy soft dels These concepts re bsc structures for mprovement of soft set theory One cn extend ths work by studyng other lgebrc structures eferences [] Molodtsov, D (999) Soft set theory-frst results Comput Mth Appl, 37, 9-37

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