Definability of band structures on posets

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 xcdot 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.