Exact-$m$-majority terms

Abstract: We say that an idempotent term $t$ is an exact-$m$-majority term if $t$ evaluates to $a$, whenever the element $a$ occurs exactly $m$ times in the arguments of $t$, and all the other arguments are equal. If $m<n$ and some variety $\mathcal V$ has an $n$-ary exact-$m$-majority term, then $\mathcal V$ is congruence modular. For certain values of $n$ and $m$, for example, $n=5$ and $m=3$, the existence of an $n$-ary exact-$m$-majority term neither implies congruence distributivity, nor congruence permutability.