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

Order Pub Date : 2022-04-20 DOI:10.1007/s11083-022-09600-y
Taiga Yoshida, Masahiko Yoshinaga
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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.

对于偏序集Q, $$-Q$$ - Q是什么?
在组合互易的背景下,对于偏序集q来说,“\(-Q\)”是什么是一个很自然的问题。在之前的工作中,基于半代数集的欧拉特征的概念,提出了“具有字典顺序的\(-Q:=Q\times \mathbb {R}\)”的定义。实际上,通过使用这个定义,Stanley对阶多项式的互易性被推广为对偏集之间递增映射的某些空间的欧拉特征的等式。本文的目的是改进这个结果,即证明如果Q的拓扑是可度量的,这些空间是同胚的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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