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

TL;DR

This work studies the Cohen–Macaulayness and well-covered properties of zero-divisor graphs associated with Boolean posets and their finite bounded product posets. It first establishes that Γ(P) is CM for Boolean posets by proving Γ(P) is well-covered and applying a constructive labeling criterion, then extends to product posets Π Pi (n≥3) with Z(Pi)={0}, showing CM holds if and only if the product is a Boolean lattice. The results yield equivalences among CM, well-coveredness, uniform 2-element sizes in the factors, and Boolean-structure, thereby generalizing known CM results for Boolean algebras to the zero-divisor graph setting. Overall, the paper connects poset Booleanity with topological-algebraic properties of corresponding graphs, enriching the classification of CM zero-divisor graphs in ordered combinatorial contexts.

Abstract

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

Figures (1)

• Figure 1: The Boolean poset $P$ and its zero-divisor graph $\Gamma(P)$.

Theorems & Definitions (33)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2: MY
• Theorem 2.3: VA
• Lemma 2.4: VA
• Theorem 3.1
• proof
• Corollary 3.2