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 $\alpha (\beta \circ \Theta )\subseteq \alpha \beta \circ \alpha \Theta \circ \alpha \beta$ 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 $R(S\circ \gamma )\subseteq RS\circ R\gamma \circ RS$ or $R(S\circ T)\subseteq RS\circ RT\circ RT\circ RS$ 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.

arXiv commentv2, entirely rewritten, the main theorems of v1 are now corollaries of more general results, v3, expanded the introduction, some further additions

References in corpus (2)

• On tractability and congruence distributivity
• On the Jónsson distributivity spectrum

Cited by in corpus (4)

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• On the number of terms witnessing congruence modularity
• The distributivity spectrum of Baker's variety
• Day's Theorem is sharp for $n$ even