Introduction to semigroup theory

The concept of \(\Gamma \)-semigroup is a generalization of semigroup. Let S and \(\Gamma \) be two nonempty sets. S is called \(\Gamma \)-semigroup if there exists a mapping \(S\times \Gamma \times S\longrightarrow S\), written as \((a, \alpha , b)\longrightarrow a\alpha b\), satisfying the identity \( (a\alpha b)\beta c\) \(=\) \(a\alpha (b\beta c) \) for all \(a, b, c\in S\) and \( \alpha , \beta \in \Gamma \). This article is a survey of some works published by different authors on \(\Gamma \)-semigroups.

\(\Gamma \)-Semigroups: A Survey

Let S = {a, b, c, ...} and = {, , , ...} be two nonempty sets. S is called a -semigroup if , for all and a, b S and , for all a, b, c S and for all , . An element is said to be an -idempotent for some if = e. A -semigroup S is called an E-inversive -semigroup if for each there exist and such that ax is a -idempotent for some . A -semigroup is called a right E--semigroup if for each -idempotent e and -idempotent f, is a -idempotent. In this paper we investigate different properties of E-inversive -semigroup and right E--semigroup.

E-Inversive Γ-Semigroups

Let S = {a, b, c, . . .} and Γ = {α, β, γ, . . . } be two nonempty sets. S is called a Γ-semigroup if aαb ∈ S, for all α ∈ Γ and a, b ∈ S and (aαb)βc = aα(bβc), for all a, b, c ∈ S and for all α, β ∈ Γ. In this paper we study the semidirect product of a semigroup and a Γ-semigroup. We also introduce the notion of wreath product of a semigroup and a Γsemigroup and investigate some interesting properties of this product.

Sumanta Chattopadhyay

Papers

Wreath product of a semigroup and a Γ-semigroup

E-Inversive Γ-Semigroups