Relation identities in 3-distributive varieties

Abstract

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

$$\begin{aligned} \alpha ( \beta \circ \Theta ) \subseteq \alpha \beta \circ \alpha \Theta \circ \alpha \beta \end{aligned}$$

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, hence do not imply the existence of a majority term. 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 remark about implication algebras.