On The Ideals of $Γ$-Semigroup
Abin Sam Tharakan, G. Sheeja
TL;DR
The paper extends ideal theory to $\\Gamma$-semigroups by defining and linking simple, $0$-simple, and completely $0$-simple structures. It develops a framework built on $0$-least $\\Gamma$-left/right ideals, primitive idempotents, and Green's relations, and proves that a $0$-simple $\\Gamma$-semigroup is completely $0$-simple iff it contains both a $0$-least left and a $0$-least right ideal, with nonzero elements forming a $D$-class and the structure being regular. It then introduces $\\Gamma$-prime ideals, provides necessary and sufficient criteria for primeness (including in the commutative case), and analyzes the behavior of unions and intersections of $\\Gamma$-prime ideals, including conditions under ACC/DCC that preserve primeness. The work culminates in a cohesive decomposition of completely $0$-simple $\\Gamma$-semigroups as unions around primitive idempotents and extends Green's relations to the $\\Gamma$-semigroup setting, offering a robust framework for further study of $\\Gamma$-ideals. These results deepen the understanding of the structure and ideal theory of $\\Gamma$-semigroups and pave the way for applications to related ternary and colored semigroup frameworks.
Abstract
The concept of $Γ$-semigroups was introduced by M. K Sen in 1981. This study aims to investigate several intriguing properties of $Γ$-semigroups and to provide the concepts of simple $Γ$-semigroups, 0-simple $Γ$-semigroups, and completely 0-simple $Γ$-semigroups. We prove that non-zero elements of the completely 0-simple $Γ$-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 $Γ$-semigroups and establish the necessary and sufficient condition for a $Γ$-semigroups to be completely 0-simple. This study further introduce $Γ$-prime ideals and gave some condition in which a $Γ$-2-sided ideal to be a $Γ$-prime. In addition, we establish a condition for a commutative $Γ$ semigroup to be $Γ$-prime. we have established how union and intersection of $Γ$-prime ideals become $Γ$-prime.