Relation identities in 3-distributive varieties

Let $\alpha$, $\beta$, $\gamma, \dots$ $\Theta$, $\Psi, \dots$ $R$, $S$, $T, \dots$ be variables for, respectively, congruences, tolerances and reflexive admissible relations. Let juxtaposition denote intersection. We show that if the identity $\alpha(\beta \circ \Theta) \subseteq \alpha \beta \circ \alpha \Theta \circ \alpha \beta$ holds in a variety $\mathcal {V}$, then $\mathcal {V}$ has a majority term, equivalently, $\mathcal {V}$ satisfies $\alpha (\beta \circ \gamma) \subseteq \alpha \beta \circ \alpha \gamma$. The result is unexpected, since in the displayed identity we have one more factor on the right and, moreover, if we let $\Theta$ be a congruence, we get a condition equivalent to $3$-distributivity, which is well-known to be strictly weaker than the existence of a majority term. The above result is optimal in many senses, for example, we show that slight variations on the displayed identity, such as $R (S \circ \gamma) \subseteq R S \circ R \gamma \circ R S$ or $R(S \circ T) \subseteq R S \circ RT \circ RT \circ RS$ hold in every $3$-distributive variety. Similar identities are valid even in varieties with $2$ Gumm terms, with no distributivity assumption. We also discuss relation identities in $n$-permutable varieties and present a few remarks about implication algebras.