Another characterization of congruence distributive varieties

Abstract

We provide a Maltsev characterization of congruence distributive varieties by showing that a variety $\mathcal {V}$ is congruence distributive if and only if the congruence identity $\alpha \cap (\beta \circ \gamma \circ \beta ) \subseteq \alpha \beta \circ \gamma \circ \alpha \beta \circ \gamma \dots $ ($k$ factors) holds in $\mathcal {V}$, for some natural number $k$.