The idempotent semigroups (bands) that give rise to partial orders by defining \(a \le b \iff a \cdot b = a\) 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 \(\{ \le \}\) 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.
