Majority term

Remarks

Example: on $\set{0, 1}$ the Boolean majority function $m(x, y, z) = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ is a majority term of the two-element Boolean algebra. Check: if two arguments equal $a$, then one of the three meets equals $a$ and the other two are at most $a$, so the join is $a$. Non-example: the term $p(x, y, z) = x - y + z$ of abelian groups satisfies $p(x, x, y) \approx y$, not $x$, so it fails the first identity. In fact no variety of groups other than the trivial variety has a majority term, since such a variety is not congruence-distributive.