Relation identities in 3-distributive varieties

Let alpha, beta, gamma, ... Theta, Psi, ... R, S, T, ... be variables for, respectively, congruences, tolerances and reflexive admissible relations. Let juxtaposition denote intersection. We show that if the identityalpha(beta omicron Theta) subset of alpha beta omicron alpha Theta omicron alpha betaholds in a variety V, then V has a majority term, equivalently, V satisfies alpha(beta omicron gamma ) subset of alpha beta omicron 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 omicron gamma) subset of RS omicron R gamma omicron RS or R(S omicron T) subset of RS omicron RT omicron RT omicron 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.