Definability of band structures on posets

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.