The associative-poset point of view on right regular bands

TL;DR

This work connects right-regular bands (RRBs) with their associative posets by exploiting the order $x\le y$ defined via $xy=x$ and constructs a left adjoint to the forgetful functor from RRBs to associative posets, valid for relational classes definable by finite conjunctions of identities. It then develops an order-theoretic approach to direct product decompositions in RRBs with a central commuting element, giving a representation lemma and a factorization theorem that classify factor congruences in terms of central-factor decompositions. The authors also examine concrete instances and show that standard SAFT methods do not apply due to the lack of a cogenerating set in the RRB category, reinforcing the need for explicit constructive adjunctions. Overall, the paper provides a conceptual bridge between poset-theoretic structure and algebraic representations of RRBs, along with tools for inner factorization and decomposition that may guide further research in this area.

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.

Figures (4)

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• Figure 3: $FU\mathbf{B}$
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Theorems & Definitions (39)

• Lemma 2.1
• Lemma 2.2
• Definition 3.1
• Definition 3.2
• Example 3.3
• Theorem 3.4
• Proposition 3.5
• Definition 3.6: $F$ on objects
• Lemma 3.7
• Lemma 3.8: $F$ on morphisms