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

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.