On the intersection ideal graph of semigroups

TL;DR

This paper studies the intersection ideal graph of semigroups, analyzing its connectivity, diameter, and other graph properties, and classifies semigroups based on these features, including automorphism groups.

Contribution

It provides a comprehensive analysis of the intersection ideal graph of semigroups, including connectivity, diameter classification, and automorphism group determination.

Findings

If the intersection ideal graph is connected, its diameter is at most 2.

Classified semigroups with diameter exactly two.

Determined automorphism groups for semigroups composed of minimal left ideals.

Abstract

The intersection ideal graph $\Gamma(S)$ of a semigroup $S$ is a simple undirected graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of $\Gamma(S)$. We show that if $\Gamma(S)$ is connected then $diam(\Gamma(S)) \leq 2$. Further we classify the semigroups such that the diameter of their intersection graph is two. Other graph invariants, namely perfectness, planarity, girth, dominance number, clique number, independence number etc. are also discussed. Finally, if $S$ is union of $n$ minimal left ideals then we obtain the automorphism group of $\Gamma(S)$.