Cohen-Macauleyness of the Zero-Divisor Graph of a Boolean Poset

TL;DR

This paper proves that the zero-divisor graph of a Boolean poset is both well-covered and Cohen–Macaulay, and characterizes when the zero-divisor graph of a product of finite bounded posets is Cohen–Macaulay.

Contribution

It establishes the Cohen–Macaulay property for zero-divisor graphs of Boolean posets and characterizes such graphs for products of finite bounded posets.

Findings

Zero-divisor graph of a Boolean poset is well-covered and Cohen–Macaulay.

For product of finite bounded posets, Cohen–Macaulayness occurs iff the poset is a Boolean lattice.

Characterization of Cohen–Macaulay zero-divisor graphs in specific poset products.

Abstract

In this paper, we prove that the zero-divisor graph $\Gamma(P)$ of a Boolean poset $P$ is both well-covered and Cohen--Macaulay. Furthermore, for a poset $\mathbf{P} = \prod_{i=1}^{n} P_i$ $(n \ge 3)$, where each $P_i$ is a finite bounded poset satisfying $Z(P_i) = \{0\}$ for all $i$, and $\le |P_1| \le |P_2| \le \cdots \le |P_n|, $ we show that the zero-divisor graph $\Gamma(\mathbf{P})$ is Cohen--Macaulay if and only if $\mathbf{P}$ is a Boolean lattice.