Complemented zero-divisor graph of posets

TL;DR

This paper characterizes when the zero-divisor graph of a poset is complemented, linking it to algebraic and topological properties, and applies these results to semigroups, rings, and vector spaces.

Contribution

It introduces new equivalent conditions for complemented zero-divisor graphs of posets and connects them to algebraic and topological frameworks.

Findings

Complemented zero-divisor graphs are characterized by equivalent algebraic and topological conditions.

The notions of complemented and uniquely complemented zero-divisor graphs coincide for posets with zero.

Applications to semigroups, rings, and vector spaces demonstrate the practical relevance of these characterizations.

Abstract

In this paper, we derive a set of equivalent conditions for the zero-divisor graph $\Gamma(Q)$ of a poset $Q$ with $0$ to be complemented, characterizing it in terms of quasi-complemented posets. Furthermore, we prove that the notions of a complemented zero-divisor graph and a uniquely complemented zero-divisor graph coincide for any poset $Q$ with $0$. In addition, we provide both algebraic and topological characterizations for $\Gamma(Q)$ to be a complemented graph. In the final section, we apply these characterizations to the zero-divisor graphs of a reduced (multiplicative) semigroup $S$ with $0$ and the comaximal (ideal) graph of an Artinian ring $R$, and the nonzero component union graph $\mathbb{UG}(\mathbb{V})$ of a finite-dimensional vector space $\mathbb{V}$ over a field $\mathbb{F}$.