On the inclusion ideal graph of semigroups

TL;DR

This paper explores the algebraic and graph-theoretic properties of the inclusion ideal graph of semigroups, analyzing its structure, invariants, and automorphisms to deepen understanding of semigroup ideals.

Contribution

It provides new insights into the structure of inclusion ideal graphs of semigroups, including conditions for connectivity, clique number, and automorphism groups, expanding the theoretical framework.

Findings

Diameter of the graph is at most 3 if connected

Characterization of the clique number in terms of minimal left ideals

Analysis of graph invariants like perfectness, planarity, girth, independence, and matching numbers

Abstract

The inclusion ideal graph $\mathcal{I}n(S)$ of a semigroup $S$ is an undirected simple graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if either $I \subset J$ or $J \subset I$. The purpose of this paper is to study algebraic properties of the semigroup $S$ as well as graph theoretic properties of $\mathcal{I}n(S)$. In this paper, we investigate the connectedness of $\mathcal{I}n(S)$. We show that diameter of $\mathcal{I}n(S)$ is at most $3$ if it is connected. We also obtain a necessary and sufficient condition of $S$ such that the clique number of $\mathcal{I}n(S)$ is $n$, where $n$ is the number of minimal left ideals of $S$. Further, various graph invariants of $\mathcal{I}n(S)$ viz. perfectness, planarity, girth etc. are discussed. For a completely simple semigroup $S$, we investigate various properties of $\mathcal{I}n(S)$ including its independence number and matching number. Finally, we obtain the automorphism group of $\mathcal{I}n(S)$.