Definability of band structures on posets
| Autor: | Kuperman, Joel, Petrovich, Alejandro, Terraf, Pedro Sánchez |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 23 pages, 10 figures. Comments are welcome. v2: As per referee's request, we are splitting this work. The first manuscript corresponds to version 2 |
| Databáze: | arXiv |
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