A Novel of Ideals and Fuzzy Ideals of Γ-Semigroups

Abstract

A Γ-semigroup is an algebraic structure considered as a generalization of a semigroup. Let S and Γ be nonempty sets. Then S is called a Γ-semigroup if there exists a mapping S × Γ × S → S defined as (a, γ, b) → aγb satisfying the axiom (aαb)βc = aα(bβc) for all a, b, c ∈ S and α, β ∈ Γ and we say that f is a fuzzy subset of a set S if f is a function from S into the closed interval [0, 1]. These two concepts are interesting to study together. In this study, we define almost quasi-Γ-ideals of a Γ-semigroup and study some properties of them such as the union of two almost quasi-Γ-ideals is an almost quasi-Γ-ideal but their intersection need not always be an almost quasi-Γ-ideal. We also define and study fuzzy almost quasi-Γ-ideals of a Γ-semigroup. We give some relationships between almost quasi-Γ-ideals and fuzzy almost quasi-Γ-ideals of Γ-semigroups. Moreover, almost bi-Γ-ideals and fuzzy almost bi-Γ-ideals of Γ-semigroups will be defined and we give properties of them. In addition, we investigate relationships between almost bi-Γ-ideals and fuzzy almost bi-Γ-ideals. Finally, we define new types of ideals, fuzzy ideals, almost ideals and fuzzy almost ideals of Γ-semigroups by using elements of Γ. We investigate properties of them and show the relationships between these ideals and their fuzzifications.