INTERVAL-VALUED INTUITIONISTIC FUZZY CLOSED IDEALS OF BG-ALGEBRA AND THEIR PRODUCTS

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1 ITEVL-VLED ITITIOISTIC FZZY CLOSED IDELS OF G-LGE D THEI PODCTS Tapan Senapat #, onoranjan howmk *, adhumangal Pal #3 # Department of ppled athematcs wth Oceanology Computer Programmng, Vdyasagar nversty, dnapore -7 0, Inda mathtapan@gmalcom 3 mmpalvu@gmalcom *Department of athematcs, V T T College, dnapore- 7 0, Inda mbvttc@gmalcom STCT In ths paper, we apply the concept of an nterval-valued ntutonstc fuzzy set to deals closed deals n G-algebras The noton of an nterval-valued ntutonstc fuzzy closed deal of a G-algebra s ntroduced, some related propertes are nvestgated lso, the product of nterval-valued nntutonstc fuzzy G-algebra s nvestgated KEYWODS D PHSES G-algebras, nterval-valued ntutonstc fuzzy sets IVIFSs), IVIF-deals, IVIFC-deals, homomorphsm, equvalence relaton, upperlower)-level cuts, product of G-algebra ITODCTIO lgebrac structures play an mportant role n mathematcs wth wde range of applcatons n many dscplnes such as theoretcal physcs, computer scences, control engneerng, nformaton scences, codng theory etc On the other h, n hlng nformaton regardng varous aspects of uncertanty, non-classcal logc a great extenson development of classcal logc) s consdered to be more powerful technque than the classcal logc one The non-classcal logc, therefore, has now a days become a useful tool n computer scence oreover, non-classcal logc deals wth the fuzzy nformaton uncertanty In 965, Zadeh [8] ntroduced the noton of a fuzzy subset of a set as a method for representng uncertanty n real physcal world Extendng the concept of fuzzy sets FSs), many scholars ntroduced varous notons of hgherorder FSs mong them, nterval-valued fuzzy sets IVFSs) provdes wth a flexble mathematcal framework to cope wth mperfect mprecse nformaton oreover, ttanssov [,6] ntroduced the concept of ntutonstc fuzzy sets IFSs) the nterval-valued ntutonstc fuzzy sets IVIFSs), as a generalzaton of an ordnary FSs In 966, Ima Isek [3] ntroduced two classes of abstract algebra: CK-algebras CIalgebras It s known that the class of CK-algebra s a proper subclass of the class of CIalgebras In [, ] Hu L ntroduced a wde class of abstract algebras: CH-algebras They have shown that the class of CI-algebra s a proper subclass of the CH-algebras eggers DOI : 05/jfls003 7

2 Km [0] ntroduced a new noton, called a -algebras whch s related to several classes of algebras of nterest such as CH/CI/CK-algebras Cho Km [8] dscussed further relatons between -algebras other topcs especally quasgroups Park Km [] shown that every quadratc -algebra on a feld X wth X 3 s a CI-algebra Jun et al [0] fuzzyfed normal) -algebras gave a characterzaton of a fuzzy -algebras Km Km [7] ntroduced the noton of G-algebras, whch s a generalzaton of -algebras hn Lee [] fuzzfed Galgebras Saed [4] ntroduced fuzzy topologcal G-algebras, nterval- valued fuzzy Galgebras In the same year Saed [3] also dscussed some results of nterval-valued fuzzy Galgebra Senapat et al [6] presented the concept basc propertes of nterval-valued ntutonstc fuzzy IVIF) G-subalgebras In ths paper, nterval-valued ntutonstc fuzzy deal IVIF-deal) of G-subalgebras s defned lot of propertes are nvestgated The noton of equvalence relatons on the famly of all nterval-valued ntutonstc fuzzy deals of a G-algebra s ntroduced nvestgated some related propertes The product of IVIF) G-subalgebra has been ntroduced some mportant propertes of t are also studed The rest of ths paper s organzed as follows The followng secton brefly revews some background on G-algebra, G-subalgebra, refnement of unt nterval, IVIF) G-subalgebras In Secton 3, the concepts operatons of IVIF-deal) nterval-valued ntutonstc fuzzy closed deal IVIFC-deal) are proposed dscuss ther propertes n detal In Secton 4, some propertes of IVIF-deals under homomorphsms are nvestgated In Secton 5, equvalence relatons on IVIF-deals s ntroduced In secton 6, product of IVIF G-subalgebra some of ts propertes are studed Fnally, n Secton 7, concluson scope of for future research are gven PELIIIES In ths secton, some defntons are recalled whch are used n the later sectons The G-algebra s a very mportant branch of a modern algebra, whch s defned by Km Km [7] Ths algebra s defned as follows Defnton [7] G-algebra) non-empty set X wth a constant 0 a bnary operaton s sad to be G-algebra f t satsfes the followng axoms x x 0 x 0 x 3 x 0 for all y X G-algebra s denoted by X,,0) n example of G-algebra s gven below Example Let X {0,,,3,4,5} be a set The bnary operaton over X s defned as * Ths table satsfes all the condtons of Defnton Hence, X,,0) s a G-algebra partal orderng on X can be defned by x y f only f x y 0 ow, we 8

3 ntroduce the concept of G-subalgebra over a crsp set X the bnary operaton n the followng Defnton [7] G-subalgebra) non-empty subset S of a G-algebra X s called a subalgebra of X f x y S, for all y S From ths defnton t s observed that, f a subset S of a G-algebra satsfes only the closer property, then S becomes a G-subalgebra Defnton 3 Ideal) non-empty subset I of a G-algebra X s called an deal of X f ) 0 I ) x y I y I x I, for any y X n deal I of a G-algebra X,,0) s called closed f 0 x I, for all x I The IVIFS s a partcular type of FS hn Lee [] extends the concepts of G-subalgebra from crsp set to fuzzy set In the fuzzy set, the membershp values of the elements are wrtten together along wth the elements The membershp values le between 0 The defnton of ths set s gven below Defnton 4 [8] Fuzzy set) Let X be the collecton of objects denoted generally by x then a fuzzy set n X s defned as {< µ >: x X} where µ s called the membershp value of x n 0 µ Combned the defnton of G-subalgebra over crsp set the dea of fuzzy set hn Lee [] defned fuzzy G-subalgebra, whch s defned below Defnton 5 [] Fuzzy G-subalgebra) Let be a fuzzy set n a G-algebra Then s called a fuzzy subalgebra of X f µ x mn{ µ, µ for all y X, where µ s the membershp value of x n Defnton 6 [9] Fuzzy G-deal) fuzzy set {< µ >: x X} n X s called a fuzzy deal of X f t satsfes ) µ 0) µ ) µ mn{ µ µ for all y X In a fuzzy set only the membershp value µ of an element x s consdered, the nonmembershp value can be taken as µ Ths value also les between 0 ut n realty ths s not true for all cases, e, the non-membershp value may be strctly less than Ths dea was frst ncorporated by ttanasov [] ntated the concept of ntutonstc fuzzy set defned below Defnton 7 [] Intutonstc fuzzy set) n ntutonstc fuzzy set over X s an object havng the form { µ, ν : x X}, where µ : X [0,] ν : X [0,], wth the condton 0 ν + ν for all x X The numbers µ ν denote, respectvely, the degree of membershp the degree of nonmembershp of the element x n the set Obvously, when ν µ for every x X, the set becomes a fuzzy set Extendng the dea of fuzzy G-subalgebra, Zar Saed [3] defned ntutonstc fuzzy G-subalgebra In ntutonstc fuzzy G-subalgebra, two condtons are to be satsfed, nstead of one condton n fuzzy G-subalgebra Defnton 8 [7]Intutonstc fuzzy G-subalgebra) n IFS { µ, ν : x X} n X s called an ntutonstc fuzzy subalgebra of X f t satsfes the followng two condtons, 9

4 µ x mn{ µ, µ ν x max{ ν, ν The people observed that the determnaton of membershp value s a dffcult task for a decson maker In [9], Zadeh defned another type of fuzzy set called nterval-valued fuzzy sets IVFSs) The membershp value of an element of ths set s not a sngle number, t s an nterval ths nterval s an subnterval of the nterval [0,] Let D[0,] be the set of a subntervals of the nterval [0,] Defnton 9 [9] IVFS) n IVFS over X s an object havng the form { : x X}, where : X D[0,], where D [0,] s the set of all sub ntervals of [0,] The nterval denotes the nterval of the degree of membershp of the element x to the set, where [ L, ] for all x X Combnng the dea of ntutonstc fuzzy set nterval-valued fuzzy sets, tanassov Gargov [3] defned a new class of fuzzy set called nterval-valued ntutonstc fuzzy sets IVIFSs) defned below Defnton 0 [3] IVIFS) n IVIFS over X s an object havng the form {, : x X}, where : X D[0,] : X D[0,], where D [0,] s the set of all subntervals of [0,] The ntervals denote the ntervals of the degree of membershp degree of non-membershp of the element x to the set, where [ L, ] [ L, ], for all x X, wth the condton 0 + lso note that [, ] [, ], where [, ] represents the complement of x n For the sake of smplcty, we shall use the symbol, ) for the IVIFS {, : x X} L The determnaton of maxmum mnmum between two real numbers s very smple, but t s not smple for two ntervals swas [7] descrbed a method to fnd max/sup mn/nf between two ntervals or a set of ntervals Defnton [7] efnement of ntervals) Consder two elements D, D D[0,] If D [ a, ] D [ a, ], then rmax D, D ) [max a, a ), max b, )] whch s b b r D b D, D ) [mn a, a), mn b, b denoted by D rmn )] whch s denoted by D r D Thus, f D [ a, b ] D[0,] for,,3,4,, then we defne rsup D ) r [ sup a ), sup b )], e, D [ a, b ] Smlarly, we rnf D ) [ nf a ), nf b )] e, r D [ a, b ] ow we call D D ff a a b b Smlarly, the relatons D D D D are defned The upper lower level of an IVIF G subalgebras s defned n the earler paper of Senapat et al [8] Defnton [8]IVIF G-subalgebras) Let, ) be an IVIFS n X, where X s a G-subalgebra, then the set s IVIF G-subalgebra over the bnary operator f t satsfes the followng condtons: GS) x rmn{, L 30

5 GS) x rmax{,, for all y X Defnton 3 [8] Let, ) s an IVIF G-subalgebra of X For [ s, s ], [ t, t ] D[0,], the set :[ s, s]) { x X [ s, s]} s called upper [ s, s] -level of L :[ t, t ]) { x X [ t, ]} s called lower t, ] -level of t [ t lso the mappng of an IVIFS s defned n [6] It has some extensve propertes n the feld of IVIF G-subalgebras Defnton 4 [8] Let f be a mappng from a set X nto a set Y Let be an IVIFS n Y Then the nverse mage of, e, f ) X, f ), f )) s the IVIFS n X wth the membershp functon non-membershp functon respectvely are gven by f ) f ) f ) f ) 3 IVIFC-IDELS OF G-LGES In ths secton, IVIF-deal IVIFC-deal of G-algebra are defned prove some propostons theorems are presented In what follows, let X denote a G-algebra unless otherwse specfed Defnton 5 n IVIFS, ) n X s called an IVIF-deal of X f t satsfes: GS3) 0) 0) GS4) rmn{ GS5) rmax{ for all y X Example Consder a G-algebra X {0,,,3} wth the followng Cayley table * Let, ) be an IVIFS n X defned as 0) ) [,], ) 3) [ m, m], 0) ) [0,0] ) 3) [ n, n], where m, ], n, n ] [0,] m + n Then, ) s an IVIF-deal of X [ m [ D closed deal of IVIF deal also be derved from the above defnton Defnton 6 n IVIFS, ) n X s called an IVIFC-deal of X f t satsfes GS4), GS5) GS6) wth 0 0, for all x X Example 3 Consder a G-algebra X {0,,,3,4,5} wth the table n Example We defne an IVIFS, ) n X by, 0) [05,07], ) ) [04,06], 3) 4) 5) [03,04], 0) [0,0], ) ) [0,04], 3) 4) 5) [04,06] y routne calculatons, one can verfy that, ) s an IVIFC-deal of X 3

6 Proposton Every IVIFC-deal s an IVIF-deal The converse of above proposton s not true n general as seen n the followng example Example 4 Consder a G-algebra X {0,,,3,4,5} wth the followng table * Let us an IVIFS, ) n X by 0) [05,07], ) [04,06], ) 3) 4) 5) [03,04], 0) [0,0], ) [0,04], ) 3) 4) 5) [04,06] We know that, ) s an IVIF-deal of X ut t s not an IVIFC-deal of X snce 0 0 Corollary Every IVIF G-subalgebra satsfyng GS4) GS5) s an IVIFC-deal Theorem Every IVIFC-deal of a G-algebra X s an IVIF G-subalgebra of X Proof: If, ) s an IVIFC-deal of X, then for any x X we have 0 0 ow x rmn{ x 0 ), 0, by GS4) rmn{, 0 rmn{,, by GS6) x rmax{ x 0 ), 0, by GS5) rmax{, 0 Hence the theorem rmax{,, by GS6) Proposton If an IVIFS, ) n X s an IVIFC-deal, then for all x X, 0) 0) Proof: Straghtforward Theorem n IVIFSs {[, ],[, ]} n X s an IVIF-deal of X ff L, L L, L are fuzzy deals of X Proof: Snce L 0) L, 0), L 0) L 0), therefore 0) 0) Let L are fuzzy deals of X Let y X Then [, ] L [ mn{, mn{ ] L L rmn{[ x ],[, ]} L rmn{ Let L are fuzzy deals of X y X Then L 3

7 [, ] L [ max{, max{ ] L L rmax{[ x ],[, ]} L rmax{ Hence, {[ L, ],[ L, ]} s an IVIF deal of X Conversely, assume that, s an IVIF deal of X For any y X, we have [, ] L rmn{ rmn{[ x ],[, ] L [mn{, mn{ ] L [ L, ] x ) rmax{ rmax{[ x ],[, ]} Thus, Hence, L L [max{, max{ ] L L mn{, mn{, L L L max{, max{ L L L L, L are fuzzy deals of X, The ntersecton of two IVIFS of X s defned by tanassov [4] as follows Defnton 7 Let be two IVIFSs on X, where { [, ],[, ] : x X} L { [ L, ],[ L, ] : x X} Then the ntersecton of s denoted by s gven by {, : x X} { [ mn, ), mn, )], L L [ max, ), max, )] : x X} L The defnton of ntersecton holds good for IVIF G subalgebras L Theorem 3 Let be two IVIF-deals of a G-algebras X Then s also an IVIF-deal of G-algebra X Proof: Let y Then y ow, 0) x rmn{, L L L L } 0) x rmn{, } lso, [ ) L, ) [mn, ),mn, ] )] L L [ mn ), )),mn ) ), ) y ) L x y ) L y x y rmn{ x, [ ) L, ) ] ))] 33

8 [max, ),max, )] L L [ max ), )),max ), ) y ) L x y ) L y ) x y rmax{ x, Hence, s also an IVIF-deal of G-algebra X Ths proves that the ntersecton of any two IVIF-deals of X s agan an IVIF-deal of X The above theorem can be generalzed as Corollary Intersecton of any famly of IVIF-deals of X s agan an IVIF-deal of X In the same way by the defnton of we can prove the followng result Corollary 3 If s an IVIF-deal of X then s also an IVIF-deal of X Lemma Let, ) be an IVIF-deal of X If x y z then rmn{, z)} rmax{, z)} Proof: Let y, z X such that x y z Then x z 0 thus rmn{ rmn{ rmn{ { x z), z)}, rmn{ rmn{ 0), z)}, rmn{, z)} rmax{ rmax{ rmax{ { x z), z)}, rmax{ rmax{ 0), z)}, rmax{, z)} Lemma Let, ) be an IVIF-deal of X If x y then e, s order-reservng s order-preservng Proof: Let y X such that x y Then x y 0 thus rmn{ rmn{ 0), rmax{ rmax{ 0), sng nducton on n by Lemma Lemma we can easly prove the followng theorem Theorem 4 If, ) s an IVIF-deal of X, then x a) a) ) an 0 for any x, a, a,, a n X, mples rmn{ a), a),, an )} rmax{ a ), a ),, a )} n ))] 34

9 Here we defne two operators on IVIFS as follows: Defnton 8 Let, ) be an IVIFS defned on X The operators are defned as, ), ) n X Theorem 5 If, ) s an IVIF-deal of a G-algebra X, then ), ), both are IVIF-deals of G-algebra X Proof: For ), t s suffcent to show that satsfes the second part of the condtons GS3) GS5) We have 0) 0) Let y X Then Hence, rmn{ rmax{ snce [,] rmax{ s an IVIF-deal of G-subalgebra X For ), t s suffcent to show that satsfes the frst part of the condtons GS3) GS4) We have 0) 0) Let y X Then Hence, rmax{ rmn{ snce [,] rmn{ s an IVIF-deal of G-subalgebra X Theorem 6 n IVIFS, ) s an IVIFC-deal of X ff the sets :[ s, s]) L :[ t, t ]) are closed deal of X for every s, s ],[ t, t ] [0,] [ D Proof: Suppose that, ) s an IVIFC-deal of X For [ s, s] D[0,], obvously, 0 x :[ s, s]), where x X Let y X be such that x y : [ s, s ]) y :[ s, s]) Then rmn{ y )} [ s, s ] Then x :[ s, s ]) Hence, :[ s, s ]) s closed deal of X For [ t, t] D[0,], obvously, 0 x L :[ t, t]) Let y X be such that x y L :[ t, t ]) y L :[ t, t ]) Then rmax{ :[ t, t [ t, t ] Then x L :[ t, t ]) Hence, L ]) s closed deal of X Conversely, assume that each non-empty level subset :[ s, s]) L :[ t, t]) are closed deals of X For any x X, let [ s, s] [ t, t] Then x :[ s, s]) x L :[ t, t]) Snce 0 x : [ s, s]) L : [ t, t]), t follows that 0 [ s, s] [ t, t], for all x X If there exst α, β X such that α) < rmn{ α β ), β )}, then by takng [ s ', s'] [ α β ) + rmn{ α ), β )}], t follows that α β : [ s', s ']) 35

10 β :[ s ', s' ]), but α :[ s ', s' ]), whch s a contradcton Hence, :[ s ', s ']) s not closed deal of X gan, f there exst γ, δ X such that γ ) > rmax{ γ δ ), δ )}, then by takng [ t ', t'] [ γ δ ) + rmax{ γ ), δ )}], t follows that γ δ : [ t', t']) δ L :[ t ', t']), but γ L :[ t ', t' ]), whch s a contradcton Hence, L :[ t ', t']) s not closed deal of X Hence,, ) s an IVIFC-deal of X snce t satsfes GS3) GS4) 4 IVESTIGTIO OF IVIF-IDELS DE HOOOPHIS In ths secton, homomorphsm of IVIF G-subalgebra s defned some results are studed Let f be a mappng from the set X nto the set Y Let be an IVIFS n Y Then the nverse mage of, s defned as f ) f ), f )) wth the membershp functon non-membershp functon respectvely are gven by f ) f ) f ) f ) It can be shown that f ) s an IVIFS Defnton 9 mappng f : X Y of G-algebra s called a G-homomorphsm f f x f f, for all y X ote that f f : X Y s a G-homomorphsm, then f 0) 0 Theorem 7 [8] Let f : X Y be a homomorphsm of G-algebras If, ) s an IVIF G-subalgebra of Y, then the premage f ) f ), f )) of under f s an IVIF G-subalgebra of X Theorem 8 Let f : X Y be a homomorphsm of G-algebras If, ) s an IVIF- deal of Y, then the premage f ) f ), f )) of under f n X s an IVIFdeal of X Proof: For all x X, f ) f ) 0) f 0)) f )0) f ) f ) 0) f 0)) f )0) gan let y X Then f ) f ) rmn{ f f ), f )} rmn{ f f )} rmn{ f f ) f ) ) f ) rmax{ f f ), f )} rmax{ f f )} rmax{ f ) f ) 36

11 Hence, f ) f ), f )) s an IVIF-deal of X Theorem 9 Let f : X Y be an epmorphsm of G-algebras Then, ) s an IVIF-deal of Y, f f ) f ), f )) of under f n X s an IVIF-deal of X Proof: For any x Y, a X such that f a) x Then f a)) f ) a) f )0) f 0)) f a)) f ) a) f )0) f 0)) 0) Let y Y Then f a) x f b) y for some a, b X Thus f a)) f ) a) rmn{ f ) a b), f ) b)} rmn{ f a b)), f b))} rmn{ f a) f b)), f b))} rmn{ x, f a)) f ) a) rmax{ f ) a b), f ) b)} rmax{ f a b)), f b))} rmax{ f a) f b)), f b))} rmax{ x, Then, ) s an IVIF-deal of Y 5 EQIVLECE ELTIOS O IVIF-IDELS Let IVIFIX) denote the famly of all nterval-valued ntutonstc fuzzy deals of X let [, ] [0,] Defne bnary relatons L on IVIFIX) as D, ) : ) : ), ) L L : ) L : ) respectvely, for, ), ) n IVIFIX) Then clearly L are equvalence relatons on IVIFIX) For any, ) IVIFI X ), let [ ] respectvely, [ ] ) denote the equvalence class of modulo L 0) respectvely, L ), denote by IVIFIX)/ respectvely, IVIFIX)/ L ) the collecton of all equvalence classes modulo respectvely, L ), e, respectvely, IVIFIX)/ : {[ ], ) IVIFI X )}, IVIFIX)/ L : {[ ], ) IVIFI X )} L These two sets are also called the quotent sets ow let T X ) denote the famly of all deals of X let [, ] D[0,] Defne mappngs f g from IVIFIX) to T X ) { φ} by f ) : ) g ) L : ), respectvely, for all, ) IVIFI X ) Then f g are 37

12 clearly well-defned Theorem 0 For any [, ] D[0,], the maps f to T X ) { φ} g are surjectve from IVIFIX) Proof: Let [, ] D[0,] ote that 0% 0, ) s n IVIFIX), where 0 are nterval-valued fuzzy sets n X defned by 0 [0,0] [,] for all x X Obvously f 0 % ) 0 : ) [0,0]:[, ]) φ L [,]:[, ]) L : ) g 0 % ) Let P φ ) IVIFI X ) For P% χ, χ ) IVIFI X ), we have f P% ) χ P : ) P g P% ) L χ P : ) P Hence f P P g are surjectve Theorem The quotent sets IVIFIX)/ IVIFIX)/ L are equpotent to T X ) { φ} for every D[0,] Proof: For D[0,] let f respectvely, g ) be a map from IVIFIX)/ respectvely, IVIFIX)/ L ) to T X ) { φ} defned by f [ ] ) f ) respectvely, g [ ] ) g ) ) for all, ) IVIFI X )} If : ) : ) L : ) L : ) for, ), ) n IVIFIX), then, ), ) L ; hence [ ] [ ] [ ] [ ] Therefore the maps f g are njectve ow let P φ ) IVIFI X ) For P % χ, χ ) IVIFI X ), we have L P L P f [ P% ] ) f P% ) χ P : ) P, g [ P% ] ) g P% ) L χ : ) P L P Fnally, for 0% 0, ) IVIFI X ) we get f [ 0 % ] ) f 0 % ) 0 : ) φ g [ 0 % ] ) g 0 % ) L : ) φ Ths shows that L the proof For any D[0,], we defne another relaton f g are surjectve Ths completes on IVIFIX) as follows:, ) : ) L : ) : ) L : ) for any, ), ) IVIFI X ) Then the relaton equvalence relaton on IVIFIX) s an Theorem For any D[0,], the maps : IVIFI X ) T X ) { φ} defned by ψ ) f ) g ) for each, ) X s surjectve Proof: Let D[0,] For 0% 0, ) IVIFI X ), ψ 0 % ) f 0 % ) g 0 % ) 0 : ) ψ L : ) φ For any H IVIFI X ), there exsts H: χ H, χ H ) IVIFI X ) such that ψ H % ) f H % ) g H% ) : ) L χ H : ) H Ths completes the proof χ H 38

13 Theorem 3 The quotent sets IVIFIX)/ are equpotent to T X ) { φ} for every D [0,] Proof: For D[0,], defne a map ψ : IVIFI X )/ T X ) { φ} by ψ [ ] ) ψ ) for all [ ] IVIFI X )/ ssume that ψ [ ] ) [ ] ) ψ for any [ ] [ ] IVIFI X )/ Then f ) g ) f ) g ), e, Hence : ) L : ) : ) L : ), ), so ] [ ] 0 : 0, ) IVIFI X ) we have [ Therefore the maps ψ are njectve ow for ψ [ 0 % ] ) ψ 0 ) 0 ) g 0 ) 0 : ) L : ) φ f If H IVIFIX ), then for H: χ H, χ H ) IVIFI X ), we obtan ψ [ H ] ) ψ H ) f H % ) g H% ) : ) Thus ψ s surjectve Ths completes the proof 6 PODCT OF IVIF G-LGE χ H L χ H : ) H In ths secton, product of IVIF G-algebra s defned some results are studed Defnton 0 Let, ), ) be two IVIFSs of X The cartesan product X X,, ) s defned by ) rmn{, ) rmax{,, where : X X D[0,] : X X D[0,] for all y X Proposton 3 Let, ), ) be IVIF-deals of X, then s an IVIF-deal of X X Proof: For any x, X X, we have )0,0) rmn{ 0), 0)} rmn{,, for all y X, ) )0,0) rmax{ 0), 0)} rmn{,, for all y X, ) Let x, x, X X Then ) x, y ) rmn{ x ), y )} rmn{ rmn{ x x), x)}, rmn{ y, }} rmn{ rmn{ x x), y }, rmn{ x), }} rmn{ ) x x, y y ), ) x, )} y 39

14 rmn{ ) x, x, ), ) x, } ) x, y ) rmax{ x ), y )} Hence, rmax{ rmax{ x x), x)}, rmax{ y, }} rmax{ rmax{ x x), y }, rmax{ x), }} rmax{ ) x x, y, ) x, } rmax{ ) x, y ) x, y )), ) x, )} y s an IVIF-deal of X X Proposton 4 Let, ), ) are IVIFC-deals of X, then s an IVIFC-deal of X X Proof: ow, )0,0) ) )0 0 rmn{ 0, 0 rmn{, ) )0,0) ) )0 0 rmax{ 0, 0 rmax{, ) Hence, s an IVIFC-deal of X X Lemma 3 If, ), ) are IVIF-deals of X, then ), ) s an IVIF-deals of X X Proof: Snce ) rmn{, That s, ) rmn{, Ths mples, rmn{, ) Therefore, ) rmax{, Hence, ), ) s an IVIF-deal of X X Lemma 4 If, ), ) are IVIF-deals of X, then ), ) s an IVIF-deal of X X Proof: Snce ) rmax{, That mples, ) rmax{, Ths s, rmax{, ) Therefore, ) rmn{, Hence, ), ) s an IVIF-deal of X X y the above two lemmas, t s not dffcult to verfy that the followng theorem s vald 40

15 Theorem 4 The IVIFSs, ), ) are IVIF-deals of X ff ), ) ), ) are IVIF-deal of X X Lemma 5 If, ), ) are IVIFC-deals of X, then ), ) s an IVIFC-deals of X X Proof: Snce, ), ) are IVIFC-deals of X,, ), ) are IVIF-deals of X Thus, s IVIF-deal of X X ow )0,0) ) ) That s, )0,0) ) ) Ths gves, )0,0) ) ) Hence, ), ) s an IVIFC-deal of X X Lemma 6 If, ), ) are IVIFC-deals of X, then ), ) s an IVIFC-deals of X X Proof: The proof s smlar to the proof of the above lemma The followng theorem follows from the above two lemmas Theorem 5, ), ) are IVIFC-deals of X ff ), ) ), ) are IVIFC-deal of X X Defnton Let, ), ) s IVIF G-subalgebra of X For [ s, s],[ t, t] D[0,], the set : [ s, s]) { X X ) y ) [ s, s]} s called upper [ s, s] -level of L : [ t, t]) { X X ) [ t, t ]} s called lower t, ]-level of [ t Theorem 6 For any IVIFS, ), ), s an IVIFC-deals of X X ff the non-empty upper [ s, s] -level cut :[ s, s]) the non-empty lower [ t, t] -level cut L :[ t, t]) are closed deals of X X for any [ s, s] t, t ] [0,] [ D Proof: Let, ), ) are IVIFC-deals of X, therefore for any x, X X, )0,0) ) ) )0,0) ) ) For [ s, s] D[0,], f ) [ s, s] That s, )0,0) ) [ s, s] Ths mples, 0,0) :[ s, s]) Let x,, x, y ) X X such that x, y ) :[ s, s]) x, y ) :[ s, s]) ow, ) rmn{ ) x, y )), ) x, y )} rmn[ s, s],[ s, s]) 4

16 [ s, s] Ths mples, :[ s, s]) Thus :[ s, s]) s closed deal of X X Smlarly, L :[ t, t]) s closed deal of X X Conversely, let x, X X such that ) [ s, s] ) y ) [ t, t ] Ths mples, :[ s, s]) L :[ t, t]) Snce 0,0) :[ s, s]) 0,0) L :[ t, t]) by defnton of closed deal) Therefore, )0,0) ) [ s, s] )0,0) ) [ t, t ] Ths gves, )0,0) ) ) )0,0) ) ) Hence, s an IVIFC-deal of X X 7 Conclusons Future Work In the present paper, we have ntroduced the concept of IVIF-deal IVIFC-deal of Galgebras are ntroduced nvestgated some of ther useful propertes The product of IVIF Gsubalgebra has been ntroduced some mportant propertes of t are also studed In our opnon, these defntons man results can be smlarly extended to some other algebrac systems such as F-algebras, lattces Le algebras It s our hope that ths work would other foundatons for further study of the theory of Galgebras The results obtaned here probably be appled n varous felds such as artfcal ntellgence, sgnal processng, multagent systems, pattern recognton, robotcs, computer networks, genetc algorthm, neural networks, expert systems, decson makng, automata theory medcal dagnoss In our future study of fuzzy structure of G-algebra, the followng topcs may be consdered: ) To fnd nterval-valued ntutonstc T,S)-fuzzy deals, where S T are gven magnable trangular norms; ) To get more results n IVIFC-deals of G-algebra ther applcatons; ) To fnd ε, ε q) -nterval-valued ntutonstc fuzzy deals of G-algebras 8 EFEECES [] SS hn HD Lee, Fuzzy subalgebras of G-algebras, Commun Korean ath Soc 9) 004), 43-5 [] KT tanassov, Intutonstc fuzzy sets, Fuzzy Sets Systems, 0 986), [3] KT tanassov G Gargo, Interval-valued ntutonstc fuzzy sets, Fuzzy Sets Systems, 3) 989), [4] KT tanassov, Operatons over nterval-valued fuzzy set, Fuzzy Sets Systems, ), [5] KT tanassov, ore on ntutonstc fuzzy sets, Fuzzy Sets Systems, 33) 989), [6] KT tanassov, ew operatons defned over the ntutonstc fuzzy sets, Fuzzy Sets Systems, 6 994), 37-4 [7] swas, osenfeld's fuzzy subgroups wth nterval valued membershp functon, Fuzzy Sets Systems, ),

17 [8] J Cho HS Km, On -algebras quasgroups, Quasgroups elated Systems 7 00), 6 [9] G Deschrjver, rthmetc operators n nterval-valued fuzzy theory, Informaton Scences, ), [0] DH Foster, Fuzzy topologcal groups, Journal of athematcal nalyss pplcatons, 67) 979), [] QP Hu X L, On CH-algebras, athematcs Semnar otes, 983), [] QP Hu X L, On proper CH-algebras, ath Japonca, ), [3] Y Ima K Isek, On axom system of propostonal calcul, XIV Proc Japan cademy, 4 966), 9- [4] Y Jun, EH oh HS Km, On fuzzy -algebras, Czech ath J, 57) 00), [5] KH Km, Intutonstc fuzzy deals of semgroups, Indan J Pure ppl ath, 334) 00), [6] K Jana Pal, Some operators defned over nterval-valued ntutonstc fuzzy sets, Fuzzy Logc Optmzaton, Edtor: S a, arosa Publshng House, ew Delh, Inda, 006), 3-6 [7] C Km HS Km, On G-algebras, Demonstrato athematca, 4 008), [8] WJ Lu, Fuzzy nvarant subgroups fuzzy deals, Fuzzy Sets Systems, 8 98), 3-39 [9] uthuraj, Srdharan P Stharselvam, Fuzzy G-deals n G-algebra, Internatonal Journal of Computer pplcatons, ) 00), 6-30 [0] J eggers HS Km, On -algebras, ath Vensk, 54 00), -9 [] HK Park HS Km, On quadratc -algebras, Quasgroups elated Systems 7 00), 67 7 [] osenfeld, Fuzzy Groups, Journal of athematcal nalyss pplcatons, 35 97), 5-57 [3] Saed, Some results on nterval-valued fuzzy G-algebra, World cademy of Scence, Engneerng Technology, 5 005), [4] Saed, Fuzzy topologcal G-algebra, Internatonal athematcal Journal, 6) 005), - 7 [5] Saed, Interval-valued fuzzy G-algebra,, Kangweon-Kyungk ath Jour, 4) 006), 03-5 [6] T Senapat, howmk Pal, Fuzzy closed deals of -algebras, Internatonal Journal of Computer Scence Engneerng Technology, 0) 0), [7] T Senapat, howmk Pal, On ntutonstc fuzzy subalgebras n G-algebras, Submtted) [8] T Senapat, howmk Pal, Interval-valued ntutonstc fuzzy G-subalgebras, Journal of Fuzzy athematcs ccepted) [9] L Zadeh, Fuzzy sets, Informaton Control, 8 965), [30] L Zadeh, The concept of a lngustc varable ts applcaton to approxmate reasonng I, Informaton Scences, 8 975), [3] L Zadeh, Toward a generalzed theory of uncertanty GT)-an outlne, Informaton Scences, 7 005), -40 [3] Zar Saed, Intutonstc fuzzy deals of G-algebras, World cademy of Scence, Engneerng Technology, 5 005),

18 uthors Tapan Senapat receved hs achelor of Scence degree wth honours n athematcs n 006 from dnapore College, Pashm ednpur, West engal, Inda aster of Scence degree n athematcs n 008 from Vdyasagar nversty, West engal, Inda Hs research nterest ncludes fuzzy sets, ntutonstc fuzzy sets, fuzzy algebra lattce valued trangular norm onoranjan howmk receved hs Sc n athematcs from Indan Insttute of Technology, Kharagpur, West engal, Inda PhD from Vdyasagar nversty, Inda n respectvely He s a faculty member of VTT College, Paschm dnapore, West engal, Inda Hs man scentfc nterest concentrates on dscrete mathematcs, fuzzy sets, ntutonstc fuzzy sets, fuzzy matrces, ntutonstc fuzzy matrces fuzzy algebra adhumangal Pal receved hs Sc from Vdyasagar nversty, Inda PhD from Indan Insttute of Technology, Kharagpur, Inda n respectvely He s engaged n research snce 99 In 996, he receved Computer Dvson ward from Insttuton of Engneers Inda), for best research work Durng 997 to 999 he was a faculty member of dnapore College snce 999 he has been at the Vdyasagar nversty, Inda Hs research nterest ncludes computatonal graph theory, parallel algorthms, data structure, combnatoral algorthms, genetc algorthms, fuzzy sets, ntutonstc fuzzy sets, fuzzy matrces, ntutonstc fuzzy matrces, fuzzy game theory fuzzy algebra 44