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

Abstract

In this paper, we prove that the zero-divisor graph $\hat{I}“(P)$ 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 $\hat{I}“(\mathbf{P})$ is Cohen--Macaulay if and only if $\mathbf{P}$ is a Boolean lattice.

A minor error in Lemma 3.5 is fixed