Fuzzy Sequence ¦£ Of Semi-groups In A Number Of Issues

Let S be a nonempty set,we define a binary operation "·" on S,if(S,·) satisfies the following two conditions:(1) for any a,b∈S,a·b∈S;(2)(a·b)·c = a·(b·c) for all a,b,c∈S,then(S,·) is called a semigroup[15].If a semigroup S with an order relation "≤" such that a≤b implies xa≤xb and ax≤bx for any x∈S,then(S,·,≤) is called an ordered semigroup[16].In 1965,Zadeh introduced the concept of a fuzzy set[18].Let S be a nonempty set,any a function f from S to the real closed interval[0,1]is called a fuzzy subset of S.In 1981,the concept ofΓ-semigroups was introduced by M.K.Sen[12].Let S andΓbe any two nonempty sets,if there exists a mapping S×Γ×S→S,written as(a,γ,b)→aγb,satisfying the following identities(aγb)μc = aγ(bμc) for all a,b,c∈S andγ,μ∈Γ,then S is called aΓ-semigroup.If aΓ-semigroup S is an ordered set such that a≤b implies aγc≤bγc and cγa≤cγb for any a,b,c∈S andγ∈Γ,then S is called an orderedΓ-semigroup[14].In this thesis,we study fundamental theory of fuzzy ordered semigroups and orderedΓ-semigroups. This thesis consists of two chapters.In chapter 1,we introduce the notion of fuzzy bi-ideals of an ordered semigroup S,then give some characterizations of fuzzy bi-ideals of ordered semigroups and discuss fuzzy bi-ideals generated by ordered fuzzy points.Finally,we characterize regular duo ordered semigroup S by ordered fuzzy points and fuzzy bi-ideals of S.In chapter 2,we introduce the concepts of B-simple sub-Γ-semigroup and ordered biideals, minimal ordered bi-ideals,kernels of an orderedΓ-semigroup S then we show that an ordered bi-ideal B of an orderedΓ-semigroup S is minimal if and only if B is B-simple and that the union of all the minimal ordered bi-ideals of an orderedΓ-semigroup S is the kernel of S.Finally,we introduce the concept of regular orderedΓ-semigroups and characterize regular orderedΓ-semigroups in terms of ordered left ideals,ordered right ideals or ordered left ideals, ordered right ideals and ordered quasi-ideals.