On Some Kinds Of Fuzzy Subset In Ordered Γ-semigroups

In this dissertation, we study left (right) semiregularity, quasi-completely reg-ularity, regularity, intra-regularity of ordered Γ-semigroups in terms of fuzzy (left,right) ideals, fuzzy interior ideals, fuzzy bi-ideals and fuzzy quasi-ideals of ordered Γ-semigroups, and give the concepts of the (∈,∈∨q)-fuzzy ordered Γ-subsemigroups,(∈,∈∨q)-fuzzy (left, right) ideals,(∈,∈∨q)-fuzzy bi-ideals,(∈,∈∨q)-fuzzy interiorideals of ordered Γ-semigroups, by which we characterize the left (right) semiregular-ity, quasi-completely regularity, regularity, intra-regularity of ordered Γ-semigroups.There are there chapters, the main results are given as following:In chapter1, we mainly give some basic concepts and symbols in this paper.In the first section of chapter2, we mainly give the basic concept of fuzzyinterior ideals, fuzzy bi-ideals and fuzzy quasi-ideals of ordered Γ-semigroups andsome of their properties. The main results are given as following:Theorem2.1.6Let S be an ordered Γ-semigroup. Then each fuzzy quasi-ideal of S is a fuzzy bi-ideal of S.Theorem2.1.7Let S be an ordered Γ-semigroup and f is a fuzzy bi-idealof S, thenIn the second section of chapter2, we mainly study the left (right) semiregular-ity, quasi-completely regularity, regularity, intra-regularity of ordered Γ-semigroups,and use diferent fuzzy ideals to describe them respectively. The main results aregiven as following: Theorem2.2.1Let S be a commutative left (right) semiregular orderedΓ-semigroup. Then generalized fuzzy bi-ideal f of S is a fuzzy ideal of S.Theorem2.2.2Let S be a left (right) semiregular ordered Γ-semigroup.Then every fuzzy left (right) of S is idempotent.Corollary2.2.1Let S be a quasi-completely regular ordered Γ-semigroup.Then every fuzzy left (right) of S is idempotent.Theorem2.2.3Ordered Γ-semigroup S is a left (right) semiregular if andonly if for every fuzzy ideal g and every fuzzy bi-ideal f of S, we haveTheorem2.2.4Ordered Γ-semigroup S is a left (right) semiregular if andonly if for any fuzzy left ideals f1, f2(fuzzy right ideals g1, g2), we haveTheorem2.2.5Let S be an ordered Γ-semigroup. Then the followingstatements are equivalent:(1) S is intra-regular;(2) For any fuzzy interior ideal f of S, we have(3) For any fuzzy quasi-ideal f of S, we have(4)Any fuzzy quasi-ideal f of S semiprime;(5)Any quasi-ideal f of S is semiprime.Theorem2.2.6Let S be an ordered Γ-semigroup, f a fuzzy set of S.Then f is a fuzzy ideal of S if and only if f is a fuzzy interior ideal of S.Theorem2.2.7Let S be an ordered Γ-semigroup, f a fuzzy set of S.Then f is a generalized fuzzy bi-ideal of S if and only if f is a fuzzy bi-idealof S.In the first section of chapter3, we mainly give the concepts of the (∈,∈∨q)-fuzzy Γ-subsemigroup,(∈,∈∨q)-fuzzy (left, right) ideal,(∈,∈∨q)-fuzzy bi-ideal, (∈,∈∨q)-fuzzy interior ideal of ordered Γ-semigroups, and investigate some of theirproperties. The main results are given as following:Theorem3.1.1A fuzzy set f of an ordered Γ-semigroup S is a (∈,∈∨q)-fuzzy Γ-subsemigroup of an ordered Γ-semigroup S if and only if for allx, y∈S, γ∈Γ,Theorem3.1.2A fuzzy set f of an ordered Γ-semigroup S is a (∈,∈∨q)-fuzzy bi-ideal of an ordered Γ-semigroup S if and only if(1) For any x, y∈S, x≤y f(x)≥min{f(y),0.5};(2) For any x, y∈S, γ∈Γ, f(xγy)≥min{f(x), f(y),0.5};(3) For any x, y, z∈S, α, β∈Γ, f(xαyβz)≥min{f(x), f(z),0.5}.Theorem3.1.3A fuzzy set f of an ordered Γ-semigroup S is a (∈,∈∨q)-fuzzy left(right) ideal of an ordered Γ-semigroup S if and only if(1) For any x, y∈S, x≤y f(x)≥min{f(y),0.5};(2) For any x, y∈S, γ∈Γ,Theorem3.1.4A fuzzy subset f of an ordered Γ-semigroup S is a(∈,∈∨q)-fuzzy left(right) ideal of an ordered Γ-semigroup S if and only if(1) For any x, y∈S, x≤y f(x)≥min{f(y),0.5};(2) For any x, y∈S, γ∈Γ, f(xγy)≥min{f(x), f(y),0.5};(3) For any x, a, y∈S, α, β∈Γ, f(xαaβy)≥min{f(a),0.5}.Theorem3.1.5Let S be an ordered Γ-semigroup. If f is an (∈,∈∨q)-fuzzyideal and an (∈,∈∨q)-fuzzy Γ-subsemigroup of S, then f is an (∈,∈∨q)-fuzzyinterior ideal of S.In the second section of chapter3, the left (right) semiregular, intra-regularordered Γ-semigroups are characterized by the lower part of some (∈,∈∨q)-fuzzyideals of ordered Γ-semigroups. The main results are given as following:Theorem3.2.1Ordered Γ-semigroup S is a right semiregular if and only if for every (∈,∈∨q)-fuzzy bi-ideal f and every (∈,∈∨q)-fuzzy ideal g of S.Theorem3.2.2Ordered Γ-semigroup S is a intra-regular if and only iffor every (∈,∈∨q)-fuzzy ideal f of S.Theorem3.2.3Ordered Γ-semigroup S is a left semiregular if and only iffor every (∈,∈∨q)-fuzzy left ideal f, every (∈,∈∨q)-fuzzy ideal g and every (∈,∈∨q)-fuzzy bi-ideal h of S.In the third section of chapter3, the quasi-completely regularity, regularity,intra-regularity of ordered Γ-semigroups are investigated by the0.5-product of some(∈,∈∨q)-fuzzy ideals of ordered Γ-semigroups. The main results are given asfollowing:Theorem3.3.1Let S be an ordered Γ-semigroup. If for every (∈,∈∨q)-fuzzy interior ideal f and every (∈,∈∨q)-fuzzy left ideal g of S, f0.5g=f∩0.5g,then S is quasi-completely regular.Theorem3.3.2Let S be an ordered Γ-semigroup. If for every (∈,∈∨q)-fuzzy left ideal f and every (∈,∈∨q)-fuzzy right ideal g of S, f0.5g=f∩0.5g,S is intra-regular.Theorem3.3.3Ordered Γ-semigroup S is a regular if and only if for every(∈,∈∨q)-fuzzy bi-ideal f, f0.5S0.5f=f.