Hoefnagel, MichaelJanelidze, ZurabRodelo, Diana2021-06-242021-06-242020-030002-5240http://hdl.handle.net/10400.1/16538For a given variety V of algebras, we define a class relation to be a binary relation R subset of S(2)which is of the form R = S-2 boolean AND K for some congruence class K on A(2), where A is an algebra in V such that S subset of A. In this paper we study the following property of V : every reflexive class relation is an equivalence relation. In particular, we obtain equivalent characterizations of this property analogous to well-known equivalent characterizations of congruence-permutable varieties. This property determines a Mal'tsev condition on the variety and in a suitable sense, it is a join of Chajda's egg-box property as well as Duda's direct decomposability of congruence classes.engClass relationsCongruence permutabilityCongruence distributivityCongruence modularityDirectly decomposable congruence classesDifunctionalityEgg-box propertyMal'tsev conditionMal'tsev varietyShifting lemmaMathematicsjournal article10.1007/s00012-020-00651-z