What is $-Q$ for a poset $Q$?

TL;DR

This paper refines the understanding of the negative of a poset in combinatorial reciprocity, showing that certain spaces of increasing maps are homeomorphic under metrizability conditions, extending Stanley's reciprocity results.

Contribution

It demonstrates that the spaces of increasing maps between posets are homeomorphic when the poset's topology is metrizable, refining previous reciprocity results.

Findings

Spaces of increasing maps are homeomorphic under metrizability.

Generalization of Stanley's reciprocity to topological posets.

Refinement of the Euler characteristic approach in combinatorial reciprocity.

Abstract

In the context of combinatorial reciprocity, it is a natural question to ask what "$-Q$" is for a poset $Q$. In a previous work, the definition "$-Q:=Q\times\mathbb{R}$ with lexicographic order" was proposed based on the notion of Euler characteristic of semialgebraic sets. In fact, by using this definition, Stanley's reciprocity for order polynomials was generalized to an equality for the Euler characteristics of certain spaces of increasing maps between posets. The purpose of this paper is to refine this result, that is, to show that these spaces are homeomorphic if the topology of $Q$ is metrizable.