On Some Study Of Fuzzy Subset InΓ- Semigroups

In this dissertation, fuzzy subsets inΓsemigroups are studied. Concepts ofgeneralized anti fuzzy subsemigroup, generalized anti fuzzy （weak） regular subsemi-group, generalized anti fuzzy n pseudo left （right, bi, interior） ideals, generalized antifuzzy quasi （semiprime, weakprime） ideals are introduced, then characterizations andrelated properties of them are discussed. In the end, some regularΓsemigroups arecharacterized by fuzzy subsets which are introduced above. There are five chapters inthis paper, the main results are given as following:In Chapter 1, we mainly give some definitions and symbols in this paper.In Chapter 2, we mainly define generalized anti fuzzy subsemigroup and gener-alized anti fuzzy （weak） regular subsemigroup, and give the properties of them. Themain results are given as following:Theorem 2.1 A fuzzy subset I in aΓsemigroup S is a generalized anti fuzzysubsemigroup of S if and only if I is an （ （?）, （?）∨（?） （α,β）） anti fuzzy subsemigroup of S.Theorem 2.2 Let I be a fuzzy subset in aΓsemigroup S, then I is an （（?）,（?）∨（?） ） anti fuzzy subsemigroup of S if and only if [I]_t is a subsemigroup of S.Theorem 2.3 A fuzzy subset I inΓsemigroup S is a generalized anti fuzzysubsemigroup of S if and only if Itis a subsemigroup of S, t∈[β,α).Theorem 2.4 Let {I_i}_i∈Kbe a family of generalized anti fuzzy subsemigroupsin aΓsemigroup S, then_i∈KI_iis a generalized anti fuzzy subsemigroup of S.(_i∈KI_i)（x） =_i∈KI_i（x）, x∈S.Theorem 2.5 Let {I_i}_i∈Kbe a family of generalized anti fuzzy subsemigroups inaΓsemigroup S, and {I_i}_i∈Kbe monotonic, then_i∈KI_iis a generalized anti fuzzysubsemigroup of S. (（?）_i∈KI_i)（x） =（?）_i∈KI_i（x）, x∈S. Theorem 2.6 Let I be a fuzzy subset in aΓsemigroup S,α,β∈[0, 1] andα>β,then I is a generalized anti fuzzy subsemigroup of S if and only if （I I）∨β≥I∧α.Theorem 2.7 A non-empty subset I in aΓsemigroup S is a subsemigroup ifand only if C_I^cis a generalized anti fuzzy subsemigroup of S.Theorem 2.8 Let I1, I2be generalized anti fuzzy subsemigroups in aΓsemigroupS, then I₁（?）βI₂is a generalized anti fuzzy subsemigroup of S.Theorem 2.9 A fuzzy subset I in aΓsemigroup S is a generalized anti fuzzysubsemigroup of S if and only ifI∩αIβI .（I∩α）（x） = I（x）∧α, x∈S.Theorem 2.10 A fuzzy subset I in aΓsemigroup S is a generalized anti fuzzyregular subsemigroup of S if and only if I_t≠φis a regular subsemigroup of S,t∈[β,α).Theorem 2.12 Let I be a generalized anti fuzzy regular subsemigroup in aΓsemigroup S, then I_tis a regular subsemigroup of S if and only if for any t∈[β,α),x∈I_t（I_t= ）, there exist an idempotent e∈I_t, such that xΓI_t′= eΓI_t, and I_tis asubsemigroup of S.Theorem 2.13 Let I be a fuzzy subset in aΓsemigroup S. The followingstatements are equivalent:（1）. I is an （ （?） , （?）∨（?） （α,β）） anti fuzzy regular subsemigroup of S;（2）. I is a generalized anti fuzzy regular subsemigroup of S.Theorem 2.16 A non-empty subset I in aΓsemigroup S is a regular sub-semigroup if and only if CIcis a generalized anti fuzzy weak regular subsemigroup ofS.Theorem 2.17 Let I be a fuzzy subset in aΓsemigroup S. The followingstatements are equivalent:（1）. I is an （ （?） , （?）∨（?） （α,β）） anti fuzzy weak regular subsemigroup of S;（2）. I is a generalized anti fuzzy weak regular subsemigroup of S;（3）. I_t<≠φis a regular subsemigroup of S, t∈[β,α). In Chapter 3, we mainly study generalized anti fuzzy n pseudo left （right, bi,interior） ideals, and give the characterizations and some properties of them. The mainresults are given as following:Theorem 3.1 Let A be a fuzzy subset in aΓsemigroup S, n be a fixed positiveinteger. The following statements are equivalent:（1）. A is an ( （?）, （?）∨（?）_α,β) anti fuzzy n pseudo right （left） ideal of S;（2）. A is a generalized anti fuzzy n pseudo right （left） ideal of S;（3）. At= is a n pseudo right （left） ideal of S, t∈[β,α).Theorem 3.2 Let A be a fuzzy subset in aΓsemigroup S, then A is a n pseudoright （left） ideal of S if and only if C_A^cis a generalized anti fuzzy n pseudo right （left）ideal of S.Theorem 3.5 Let B be a fuzzy subset in aΓsemigroup S, n be a fixed positiveinteger. B is an （ （?） , （?）∨（?） （α,β）） anti fuzzy n pseudo bi-ideal of S if and only if B is ageneralized anti fuzzy n pseudo bi-ideal of S.Theorem 3.6 Let B be a fuzzy subset in aΓsemigroup S, n be a fixed positiveinteger. B is a generalized anti fuzzy n pseudo bi-ideal of S if and only if Bt= is an pseudo bi-ideal of S, t∈[β,α).Theorem 3.7 Let B be a fuzzy subset in aΓsemigroup S, then B is a n pseudobi-ideal of S if and only if CBcis a generalized anti fuzzy n pseudo bi-ideal of S.Theorem3.8 Let A be a fuzzy subset in aΓsemigroup S, n be a fixed positiveinteger. A is an （ ,∈∨q （α,β）） anti fuzzy n pseudo interior ideal of S if and only ifA is a generalized anti fuzzy n pseudo interior ideal of S.Corollary 3.4 A is an （∈,∈∨q ） anti fuzzy interior ideal of aΓsemigroup Sif and only if for any x, y, z∈S,γ,μ∈Γ, satisfyA（yγxμz）≤A（x）∨0.5.Theorem3.9 Let A be a fuzzy subset in aΓsemigroup S, n be a fixed positiveinteger. A is a generalized anti fuzzy n pseudo interior ideal of S if and only if At= is a n pseudo interior ideal of S, t∈[β,α).Theorem 3.10 Let A be a fuzzy subset in aΓsemigroup S, then A is a n pseudo interior ideal of S if and only if CAcis a generalized anti fuzzy n pseudo interior idealof S.Theorem 3.12 Let B be a n pseudo interior ideal in aΓsemigroup S, A be afuzzy subset of S. if satisfy :（1）. A（x） =α, for any x∈S \ B;（2）. A（x）≤β, for any x∈B;（3）. 2βα≤0;then A is an （q （α,β）, q （α,β）） anti fuzzy n pseudo interior ideal of S.In Chapter 4, we mainly discuss generalized anti fuzzy quasi ideals, generalizedanti fuzzy semiprime ideals, generalized anti fuzzy weakprime ideals, （∈,∈∨q ） antifuzzy prime ideals, （∈,∈∨q ） anti fuzzy maximal ideals. The main results are givenas following:Theorem 4.1 Let Q be a fuzzy subset in aΓsemigroup S, The followingstatements are equivalent:（1）. Q is a generalized anti fuzzy quasi ideal of S;（2）. Q is an （∈,∈∨q （α,β）） anti fuzzy quasi ideal of S;（3）. Qt= is a quasi ideal of S, t∈[β,α).Corollary 4.1 Q is an （∈,∈∨q ） anti fuzzy quasi ideal of aΓsemigroup S ifand only if for any x∈S, satisfyQ（x）≤（Q S）（x）∨（S Q）（x）∨0.5.Theorem 4.2 A non-empty subset Q in aΓsemigroup S is a quasi ideal if andonly if CQcis a generalized anti fuzzy quasi ideal of S.Theorem 4.3 Q is a generalized anti fuzzy quasi ideal of aΓsemigroup S, thenQα<= is a quasi ideal of S.Theorem 4.4 Let {Qi}i∈Kbe a family of generalized anti fuzzy quasi ideals in aΓsemigroup S, theni∈KQiis a generalized anti fuzzy quasi ideal of S.Theorem 4.5 Let {Qi}i∈Kbe a family of generalized anti fuzzy quasi ideals in aΓsemigroup S, and {Qi}i∈Kbe monotonic, theni∈KQiis a generalized anti fuzzyquasi ideal of S. Theorem 4.6 Let Q be a fuzzy subset in aΓsemigroup S,α,β∈[0, 1] andβ<α, then Q is a generalized anti fuzzy quasi ideal of S if and only if Q∧α≤（Q Q）∨β.Theorem 4.7 Let Q1, Q2be generalized anti fuzzy quasi ideals in aΓsemigroupS, then Q1∪0.5Q2is a generalized anti fuzzy quasi ideal of S.Theorem 4.8 Let P be a fuzzy subset in aΓsemigroup S, then P is a generalizedanti fuzzy weakprime ideal of S if and only if P is an （∈,∈∨q （α,β）） anti fuzzyweakprime ideal of S.Corollary 4.2 Let P be a fuzzy subset in aΓsemigroup S, then P is a gener-alized anti fuzzy semiprime ideal of S if and only if P is an （∈,∈∨q （α,β）） anti fuzzysemiprime ideal of S.Theorem 4.9 Let P be a fuzzy subset in aΓsemigroup S, then P is a generalizedanti fuzzy weakprime ideal of S if and only if Pt= is a prime ideal of S, t∈[β,α).Theorem 4.12 Let {Pi}i∈Kbe a family of generalized anti fuzzy weakprime idealsin aΓsemigroup S, and {Pi}i∈Kbe monotonic, theni∈KPiis a generalized antifuzzy weakprime ideal of S.Theorem 4.13 Let {Pi}i∈Kbe a family of generalized anti fuzzy weakprime idealsin aΓsemigroup S, theni∈KPiis a generalized anti fuzzy weakprime ideal of S.Theorem 4.14 Let P be a anti fuzzy prime ideal in aΓsemigroup S, then Pis a generalized anti fuzzy weakprime ideal of S if and only if P∨β≥（P P ）∧α.Theorem 4.15 Let P be an （∈,∈∨q ） anti fuzzy ideal in aΓsemigroup S,The following statements are equivalent:（1）. P is an （∈,∈∨q ） anti fuzzy weakprime ideal of S;（2）. for any fuzzy subset P1, P2of S, if P1 P2 P , then P1∨q P or P2∨q P .Theorem 4.21 Let P be an （∈,∈∨q ） anti fuzzy ideal in aΓsemigroup S,The following statements are equivalent:（1）. P is an （∈,∈∨q ） anti fuzzy maximal ideal of S;（2）. Pt= is a maximal ideal of S, t∈[0.5, 1);（3）. [P ]t= is a maximal ideal of S, t∈[0, 1).In Chapter 5, we mainly characterize some regularΓsemigroups. The main results are given as following:Theorem 5.1 The following statements in aΓsemigroup S are equivalent:（1）. S is regular;（2）. （R L）∧α≤（R∨L）∨β, in which R is a generalized anti fuzzy right idealof S, L is a generalized anti fuzzy left ideal of S.Theorem 5.2 The following statements in aΓsemigroup S are equivalent:（1）. S is regular;（2）. （A S A）∧α≤A∨β, in which A is a generalized anti fuzzy bi-ideal of S;（3）. （A S A）∧α≤A∨β, in which A is a generalized anti fuzzy quasi ideal of S.Theorem 5.3 The following statements in aΓsemigroup S are equivalent:（1）. S is regular;（2）. （A B A）∧α≤（A∨B）∨β, in which A is a generalized anti fuzzy quasiideal of S, and B is a generalized anti fuzzy ideal of S;（3）. （A B A）∧α≤（A∨B）∨β, in which A is a generalized anti fuzzy quasiideal of S, and B is a generalized anti fuzzy interior ideal of S;（4）. （A B A）∧α≤（A∨B）∨β, in which A is a generalized anti fuzzy quasiideal of S, and B is a generalized anti fuzzy bi-ideal of S;（5）. （A B A）∧α≤（A∨B）∨β, in which A is a generalized anti fuzzy bi-idealof S, and B is a generalized anti fuzzy interior ideal of S;（6）. （A B A）∧α≤（A∨B）∨β, in which A is a generalized anti fuzzy bi-idealof S, and B is a generalized anti fuzzy ideal of S;（7）. （A B A）∧α≤（A∨B）∨β, in which A is a generalized anti fuzzy generalizedbi-ideal of S, and B is a generalized anti fuzzy interior ideal of S.Theorem 5.4 Let S be aΓsemigroup. The following statements are equivalent:（1）. S is semisimple;（2）. （A A）∧α≤A∨β, for any generalized anti fuzzy ideal A of S;（3）. （A1 A2）∧α≤（A1∪A2）∨β, for any generalized anti fuzzy ideal A1, A2of S.Theorem 5.7 Let S be aΓsemigroup, then S is cycle if and only if a generalizedanti fuzzy subsemigroup of S is a generalized anti fuzzy Drazin subsemigroup of S. Theorem 5.8 AΓsemigroup S is leftπregular if and only if for any x∈S,there exist n∈Z+, such thatL（xγ1xγ2γn 1x）∧α≤L（xμ1xμ2μnx）∨β,for any generalized anti fuzzy left ideal L of S, andγ1,γ2, ,γn 1,μ1,μ2, ,μn∈Γ.Theorem 5.12 The following statements in aΓsemigroup S are equivalent:（1）. S is regular;（2）. （R L）+= （R∨L）+, in which R is an （∈,∈∨q ） anti fuzzy right ideal of S,and L is an （∈,∈∨q ） anti fuzzy left ideal of S.Theorem 5.13 The following statements in aΓsemigroup S are equivalent:（1）. S is regular;（2）. （R∨B∨L）+≥（R B L）+, in which R is an （∈,∈∨q ） anti fuzzy rightideal of S, and L is an （∈,∈∨q ） anti fuzzy left ideal of S, and B is an （∈,∈∨q ） anti fuzzy generalized bi-ideal of S;（3）. （R∨B∨L）+≥（R B L）+, in which R is an （∈,∈∨q ） anti fuzzy rightideal of S, and L is an （∈,∈∨q ） anti fuzzy left ideal of S, and B is an （∈,∈∨q ） anti fuzzy bi-ideal of S;（4）. （R∨B∨L）+≥（R B L）+, in which R is an （∈,∈∨q ） anti fuzzy rightideal of S, and L is an （∈,∈∨q ） anti fuzzy left ideal of S, and B is an （∈,∈∨q ） anti fuzzy quasi ideal of S.