Definability of band structures on posets

Abstract

The idempotent semigroups (bands) that give rise to partial orders by defining $a\le b⟺a\cdot b=a$ 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 $\{\le \}$ is not first-order axiomatizable. We also show that the Axiom of Choice is equivalent over $ZF$ 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.

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