On Γ-Semigroups Containing Two-sided Bases

Abstract

Let 𝑆 = {𝑥, 𝑦, 𝑧, … } and Γ = {𝛼, 𝛽, 𝛾, … } be any two non-empty sets. We call 𝑆 a Γ-semigroup if 𝑆 satisfies
the following two conditions:
(i) 𝑥𝛼𝑦 ∈ 𝑆 for all 𝑎, 𝑏 ∈ 𝑆 and 𝛼 ∈ Γ;
(ii) (𝑥𝛼𝑦)𝛽𝑧 = 𝑥𝛼(𝑦𝛽𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝑆 and 𝛼, 𝛽 ∈ Γ.
It is observed that every semigroup is a Γ-semigroup. The main purpose of this paper, based on the notion of two-sided ideals generated by non-empty sets, we introduce the notion of two-sided bases of a Γsemigroup. We characterize when a non-empty subset of a Γ-semigroup is a two-sided base, prove that for any two two-sided bases of a Γ-semigroup have the same cardinal numbers and study the algebraic structure of a Γsemigroup containing two-sided bases.