Some Study Of The Fuzzy Theory Of Γ-semigroups

In this dissertation, the fuzzy congruences and intuitionistic fuzzy congruences on Γ-semigroups are studied. Concepts of the fuzzy congruences, intuitionistic fuzzy congruences and its properties are given. We also give the largest fuzzy con-gruence (intuitionistic fuzzy congruence) contained in fuzzy equivalence μ(intuitionistic fuzzy equivalence R) and the smallest fuzzy congruence (intuitionistic fuzzy congru-ence) containing fuzzy relation θ (intuitionistic fuzzy relation R) on a Γ-semigroup. Then, we give some properties about fuzzy congruence classes and intuitionistic fuzzy congruence classes on a Γ-semigroup. There are three chapters in this pa-per, the main contents are given as following.In chapter1, we give some basic concepts.In chapter2, we give some properties about the fuzzy congruences on a Γ-semigroup. The main results are given as following:Theorem2.1Let μ be a fuzzy equivalence on a Γ-semigroup S. Then μ is a fuzzy congruence on S if and only if μ is a fuzzy left and right compatible relation on S.Theorem2.2If μ is an equivalence (congruence) on a Γ-semigroup S, then λμ is a fuzzy equivalence (fuzzy congruence) on STheroem2.3Let μ, v be two fuzzy congruence on a Γ-semigroup S, then the following conditions are equivalent:(1)μ o v is a fuzzy congruence on S;(2)μ o v is a fuzzy equivalence on S;(3)μo v=v oμ. Theroem2.5Let S be a Γ-semigroup,ρ∈FR(S), thenρ is a fuzzy congru-ence on S if and only if for every∈6[0.1].ρtis a congruence on S.Theroem2.6Let μ,v be two fuzzy equivalences on a Γ-semigroup S, then(1) μ*(?)μ;(2)(μ*)-1=(μ-1)*;(3) μ(?)v(?)μ*(?)v*;(4)(μ∩v)*=μ*∩v*;(5) μ=μ*(?)μ. is a fuzzy left and right compatible relation.Theroem2.7Let μ be a fuzzy equivalence on a Γ-semigroup S, then μ*is the largest fuzzy congruence on S contained in μ.Theroem2.8Let S be a Γ-semigroup, ρ, θ∈FR(S). Then the following con-ditions hold:(1) θc(?)θ;(2)(θc)-1=(θ-1)c(3) ρ(?)θ(?)ρc(?)θc;(4)(ρ∩θ)c=ρc∩θ*;(5)(θc)c=θC:(6) θ=θC (?)θ is a fuzzy left and right compatible relation;(7)△cS=△S.Theorem2.9Let S be a Γ-semigroup, θ∈FR(S), then θ#=t(θc∪(θ-1)c∪△S).Theorem2.10Let μ be a fuzzy equivalence on a Γ-semigroup S, then for every μa,μb∈S/μ,μaa=μb(?)μ(a,b)=1.Theroem2.11Let S be a Γ-semigroup, μ is a fuzzy congruence on S. then for every μa,μb∈S/μ,α∈Γ, we have μa oμb(?)μaab.Theorem2.13Let S be a regular Γ-semigroup. μ is a fuzzy congruence on S. then the following conditions equivalent:(1) μa is an idempotent of S/μ, (2)μa=μe for some e∈E E(S).Theorem2.14Let S be an inverse Γ-semigroup;μ is a fuzzy congruence on S and a∈S. If there exists an idempotent f in S such that μ(a,f)>0, then there exists an idempotent e in S such that μ(a, e)≥μ(a,f).In chapter3, we give some properties about the intuitionistic fuzzy congru-ences on a Γ-semigroup. The main results are given as following:Theorem3.1Let R=[μR,VR] be an intuitionistic fuzzy equivalence on a Γ-semigroup S, then R is an intuitionistic fuzzy congruence on S if and only if R is an intuitionistic fuzzy left and right compatible relation on STheorem3.2Let R, Q be two intuitionistic fuzzy congruences on a Γ-semigroup S. Then the following conditions equivalent:(1) R o Q is an intuitionistic fuzzy congruence on S(2) R o Q is an intuitionistic fuzzy equivalence on S(3) R o Q is an intuitionistic fuzzy symmetry relation(4) R o Q=Q o R.Theroem3.3Let R, Q be two intuitionistic fuzzy equivalences on a Γ-semigroup S. Then the following conditions hold:(1)R°(?)R;(2)(R°)-1=(R-1)°(3) R(?)Q(?)R°(?)Q°;(4)(R∩Q)°=R°∩Q°;(5)R=R°(?) R is an intuitionistic fuzzy left and right compatible relation.Theroem3.4Let R be an intuitionistic fuzzy equivalence on a Γ-semigroup S, then R°is the largest intuitionistic fuzzy congruence on S contained in R.Theroem3.5Let S be a Γ-semigroup, R,Q∈IFR(S). Then the following conditions hold:(1) R(?)R; (2)(Rc)-1=(R-1)c(3) R(?)Q(?)Rc(?)Qc;(4)(R∪Q)c=Rc∪Qc(5)(Rc)c=Rc:(6) R=Rc(?) R is an intuitionistic fuzzy left and right compatible relation;(7)△c=△,▽C=▽.Theorem3.6Let S be a Γ-semigroup, R∈IFR(S), the R#=t(Rc∪(R-1)c∪△).Theorem3.7Let S be a regular Γ-semigroup, R is an intuitionistic fuzzy congruence on S. Then the following conditions equivalent:(1) Ra is an idempotent in S/R:(2) Re=Ra for some e∈E(S)Theorem3.8Let S be an inverse Γ-semigroup; R is an intuitionistic fuzzy congruence on S and a∈S. If there exists an idempotent f in S such that μR(a,f)>0,vR(a,f)<0,then there exists an idempotent e in S such that μR(a,e)≥μR(a,f),vR(a,e)≤vR(a,f).