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# Exact-$m$-majority terms

Sep 2022

We say that a 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 $t$, andall the other arguments are equal. If $m<n$ and some variety $\mathcal V$ hasan $n$-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 existenceof an $n$-ary exact-$m$-majority term neither implies congruencedistributivity, nor congruence permutability.

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