The associative-poset point of view on right regular bands

TL;DR

This paper explores the relationship between right regular bands and their associated posets, introducing a construction of a left adjoint functor and extending representation results to certain RRBs.

Contribution

It constructs a left adjoint to the forgetful functor from RRBs to posets and generalizes inner representations of semilattice decompositions to RRBs with commuting elements.

Findings

Constructed a left adjoint functor for RRBs and posets.

Generalized semilattice decomposition representations to RRBs.

Established a connection between RRBs and associative posets.

Abstract

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 $(P,\leq)$. 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.