UNIONS OF ADMISSIBLE RELATIONS AND CONGRUENCE DISTRIBUTIVITY

We show that a variety V is congruence distributive if and only if there is some h such that the inclusion (0.1) Θ ∩ (σ ◦ σ) ⊆ (Θ ∩ σ) ◦ (Θ ∩ σ) ◦ . . . (h factors) holds in every algebra in 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 (0.1). It is an open problem whether (0.1) 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 for- mulations for (0.1) 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.