On the diameter and girth of zero-divisor graphs of inverse semigroups

TL;DR

This paper characterizes the diameter and girth of zero-divisor graphs of inverse semigroups, classifies zero-free inverse semigroups via group congruences, and relates graph inverse semigroups to underlying graph structures.

Contribution

It provides a complete characterization of zero-divisor graph properties for inverse semigroups and links these properties to algebraic and combinatorial structures.

Findings

Zero-divisor graphs of inverse semigroups are characterized by their diameter and girth.

Inverse semigroups without zero are classified using the least group congruence.

Diameter and girth of graph inverse semigroups are described in terms of the underlying graph.

Abstract

Let $S$ be an inverse semigroup with zero and let $Z(S)^\times$ be its set of non-zero divisors with respect to the natural partial order $\le $ on $S$, that is, $a \in Z(S)^\times $ if there exists $b\in S\setminus\{0\}$ with $\omega(a, b) = \{c \in S: c \leq a\ \mbox{and}\ c \leq b\}=\{0\}$. The set $Z(S)^\times$ makes up the vertices of the corresponding {\it zero-divisor graph} $\Gamma (S)$, with two distinct vertices $a, b$ forming an edge if $\omega(a, b)=\{0\}$. We characterize {\it zero-divisor graphs} of inverse semigroups in terms of their diameter and girth. We also classify inverse semigroups without zero by building a connection between the diameter (girth) and the least group congruence $\sigma$ on an inverse semigroup without zero. Finally, we give a description of the diameter and girth of graph inverse semigoups $I(G)$ in terms of the set of vertices and the set of edges of a graph $G$.