The associative-poset point of view on right regular bands

We present two results on the relation between the class of right regular bands (RRBs) and their underlying associative posets. The first one is a construction of a left adjoint to the forgetful functor that takes an RRB $(P,\cdot )$ to the corresponding $(P,\le )$. The construction of such a left adjoint is actually done in general for any class of relational structures $(X,R)$ obtained from a variety, where $R$ is defined by a finite conjunction of identities. The second result generalizes the “inner” representations of direct product decompositions of semilattices studied by the second author to RRBs having at least one commuting element.