Abstract
Ideal theory plays an important role in studying in many algebraic structures, for example, rings, semigroups, semirings, etc. The algebraic structure Г-semigroup is a generalization of the classical semigroup. Many results in semigroups were extended to results in Г-semigroups. Many results in ideal theory of Г-semigroups were widely investigated. In this paper, we first focus to study some novel ideals of Г-semigroups. In Section 2, we define almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups by using the concept ideas of interior Г-ideals and almost Г-ideals of Г-semigroups. Every almost interior Г-ideal of a Г-semigroup S is clearly a weakly almost interior Г-ideal of S but the converse is not true in general. The notions of both almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups are generalizations of the notion of interior Г-ideal of a Г-semigroup S. We investigate basic properties of both almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups. The notion of fuzzy sets was introduced by Zadeh in 1965. Fuzzy set is an extension of the classical notion of sets. Fuzzy sets are somewhat like sets whose elements have degrees of membership. In the remainder of this paper, we focus on studying some novelties of fuzzy ideals in Г-semigroups. In Section 3, we introduce fuzzy almost interior Г-ideals and fuzzy weakly almost interior Г-ideals of Г-semigroups. We investigate their properties. Finally, we give some relationship between almost interior Г-ideals [weakly almost interior Г-ideals] and fuzzy almost interior Г-ideals [fuzzy weakly almost interior Г-ideals] of Г-semigroups.
Highlights
Introduction and preliminariesIn 1965, Zadeh [25] first introduced the notion of fuzzy subsets
We investigate basic properties of both almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups
We give some relationship between almost interior Γ-ideals [weakly almost interior Γ-ideals] and fuzzy almost interior Γ-ideals [fuzzy weakly almost interior Γ-ideals] of Γ-semigroups
Summary
In 1965, Zadeh [25] first introduced the notion of fuzzy subsets. Applications of fuzzy subsets have been developed in many fields. Wattanatripop and Chinram [19] using the concepts of quasi Γ-ideals in Γ-semigroups and almost Γ-ideals in Γ-semigroups, studied the basic properties of both almost quasi Γ-ideals and fuzzy almost quasi Γ-ideals of Γsemigroups. They gave the remarkable relationship between almost quasi-Γ-ideals and their fuzzification. Simuen, Abdullah, Yonthanthum and Chinram [18] introduced the concepts of almost bi Γ-ideals and fuzzy almost bi-Γ-ideals of Γ-semigroups They gave some properties and investigated relationship between almost bi Γ-ideals of Γsemigroups and their fuzzification. We define the fuzzifications of almost interior Γ-ideals [weakly almost interior Γ-ideals] of Γ-semigroups and give some relationship between almost interior Γ-ideals [weakly almost interior Γ-ideals] and their fuzzification
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