On The Ideals of $\Gamma$-Semigroup

TL;DR

This paper explores the properties and classifications of $\Gamma$-semigroups, introducing concepts like simple and 0-simple structures, and characterizes their ideals and regularity conditions.

Contribution

It provides new characterizations of $\Gamma$-semigroups, including the concepts of simple, 0-simple, and $\Gamma$-prime ideals, with necessary and sufficient conditions for these structures.

Findings

Non-zero elements of completely 0-simple $\Gamma$-semigroups form a D-class and are regular.

Conditions for $\Gamma$-prime ideals and their unions and intersections are established.

Characterizations of commutative $\Gamma$-semigroups as $\Gamma$-prime are provided.

Abstract

The concept of $\Gamma$-semigroups was introduced by M. K Sen in 1981. This study aims to investigate several intriguing properties of $\Gamma$-semigroups and to provide the concepts of simple $\Gamma$-semigroups, 0-simple $\Gamma$-semigroups, and completely 0-simple $\Gamma$-semigroups. We prove that non-zero elements of the completely 0-simple $\Gamma$-semigroups form a D-class and are regular. Fundamental elements of these structures are explored, and we provide concrete results that characterize them using various ideals of $\Gamma$-semigroups and establish the necessary and sufficient condition for a $\Gamma$-semigroups to be completely 0-simple. This study further introduce $\Gamma$-prime ideals and gave some condition in which a $\Gamma$-2-sided ideal to be a $\Gamma$-prime. In addition, we establish a condition for a commutative $\Gamma$ semigroup to be $\Gamma$-prime. we have established how union and intersection of $\Gamma$-prime ideals become $\Gamma$-prime.