Unions of admissible relations
Paolo Lipparini · arXiv (Cornell University) · 2017
We show that a variety $\mathcal V$ is congruence distributive if and only if there is some $h$ such that the inclusion (1) $Θ\cap ( σ\circ σ) \subseteq ( Θ\cap σ) \circ ( Θ\cap σ) \circ \dots $ ($h$ factors) holds in every algebra in $\mathcal V$, for every tolerance $Θ$ and every U-admissible relation $σ$. 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 $Θ$ 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.