Unions of admissible relations

TL;DR

This paper characterizes congruence distributivity in varieties using unions of admissible relations and explores related algebraic identities, providing a Maltsev-type characterization and open problems for further research.

Contribution

It introduces a new characterization of congruence distributivity via unions of admissible relations and offers a Maltsev-type characterization for a key inclusion.

Findings

Characterization of congruence distributivity using unions of admissible relations

Maltsev-type characterization for the inclusion involving U-admissible relations

Open problem on the equivalence when $ heta$ is a U-admissible relation

Abstract

We show that a variety $\mathcal V$ is congruence distributive if and only if there is some $h$ such that the inclusion (1) $\Theta \cap ( \sigma \circ \sigma ) \subseteq ( \Theta \cap \sigma ) \circ ( \Theta \cap \sigma ) \circ \dots $ ($h$ factors) holds in every algebra in $\mathcal V$, for every tolerance $\Theta$ and every U-admissible relation $\sigma$. By a U-admissible relation we mean a binary relation which is the set-theoretical union of a set of reflexive and admissible relations. For any fixed $h$, a Maltsev-type characterization is given for the inclusion (1). It is an open problem whether (1) is still equivalent to congruence distributivity when $\Theta$ is assumed to be a $U$-admissible relation, rather than a tolerance. In both cases many equivalent formulations for (1) are presented. The results suggest that it might be interesting to study the structure of the set of U-admissible relations on an algebra, as well as identities dealing with such relations.