Definability of band structures on posets

Joel Kuperman; Alejandro Petrovich; Pedro Sánchez Terraf

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arXiv · 2404.07877

Definability of band structures on posets

Joel Kuperman·Alejandro Petrovich·Pedro Sánchez Terraf

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Abstract

The idempotent semigroups (bands) that give rise to partial orders by defining $a \leq b \iff a \cdot b = a$ are the "right-regular" bands (RRB), which are axiomatized by $x\cdot y \cdot x = y \cdot x$. In this work we consider the class of "associative posets", which comprises all partial orders underlying right-regular bands, and study to what extent the ordering determines the possible "compatible" band structures and their canonicity. We show that the class of associative posets in the signature $\{ \leq \}$ is not first-order axiomatizable. We also show that the Axiom of Choice is equivalent over $\mathit{ZF}$ to the fact that every tree with finite branches is associative. We study the smaller class of "normal" posets (corresponding to right-normal bands) and give a structural characterization.

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Joel Kuperman, Alejandro Petrovich, Pedro Sánchez Terraf. 2024-04-11. Definability of band structures on posets. https://arxiv.org/abs/2404.07877

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