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What is \(-Q\) for a poset Q?

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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.

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Acknowledgements

Masahiko Yoshinaga was partially supported by JSPS KAKENHI Grant Numbers JP19K21826, JP18H01115. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Hokkaido University, North 10, West 8, Kita-ku, Sapporo, 060-0810, Japan

    Taiga Yoshida

  2. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Masahiko Yoshinaga

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  1. Taiga Yoshida
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  2. Masahiko Yoshinaga
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Correspondence to Masahiko Yoshinaga.

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Yoshida, T., Yoshinaga, M. What is \(-Q\) for a poset Q?. Order 40, 149–155 (2023). https://doi.org/10.1007/s11083-022-09600-y

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  • DOI: https://doi.org/10.1007/s11083-022-09600-y

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