Exact-$m$-majority terms

TL;DR

This paper introduces the concept of exact-$m$-majority terms in algebraic structures and explores their implications for the properties of varieties, such as congruence modularity, distributivity, and permutability.

Contribution

It defines exact-$m$-majority terms and investigates their role in characterizing algebraic varieties, revealing that certain types imply congruence modularity but not distributivity or permutability.

Findings

Existence of exact-$m$-majority terms implies congruence modularity.

For specific $n$ and $m$, such terms do not imply distributivity or permutability.

The concept helps distinguish between different algebraic property implications.

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.