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arXiv · 2501.13408

On The Ideals of $\Gamma$-Semigroup

Abin Sam Tharakan·G. Sheeja
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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.

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Abin Sam Tharakan, G. Sheeja. 2025-01-23. On The Ideals of $\Gamma$-Semigroup. https://arxiv.org/abs/2501.13408

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