A formula for the Euler characteristic of a poset through the determinant of the order-complement matrix

TL;DR

This paper introduces the order-complement matrix $\overline{\mathbf Z}=\mathbf J-\mathbf Z$ of a finite poset and proves a closed formula for its characteristic polynomial, along with a determinant relation $\det(\overline{\mathbf Z}) = (-1)^{n+1} \tilde{\chi}(P)$. The approach leverages linear-extension-induced upper unitriangular form of the zeta matrix $\mathbf Z$ and a Neumann-series inverse to derive an explicit polynomial $p(\lambda)=\det(\overline{\mathbf Z}-\lambda I)$, from which the determinant formula follows. This provides a new linear-algebraic expression for the reduced Euler characteristic of a poset and situates the result within the broader context of determinant formulas and Euler characteristics in poset and category theory. The paper also outlines directions for future work on the spectral properties of $\overline{\mathbf Z}$ and their connections to combinatorial invariants.

Abstract

Given a finite poset $P$, its zeta matrix $\mathbf Z$ encode fundamental incidence-theoretic information about the order structure. In this paper we introduce and study the \emph{order-complement matrix} $\overline{\mathbf Z} = \mathbf J - \mathbf Z$, where $\mathbf J$ is the all-ones matrix. We prove a closed formula for its characteristic polynomial and for its determinant, showing that $\det(\overline{\mathbf Z}) = (-1)^n \tildeχ(P)$, where $n = |P|$ and $\tildeχ(P)$ is the reduced Euler characteristic of $P$. This provides a new, unexpectedly simple linear-algebraic expression for the Euler characteristic of a poset, complementing existing determinant formulas for matrices derived from incidence relations.

Theorems & Definitions (7)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 1
• proof