Arities and aritizabilities of group, monoid and groupoid theories

TL;DR

This paper investigates the conditions under which group, monoid, and groupoid theories are aritizable, revealing that group theories are aritizable only if the group is finite, unlike monoids and groupoids.

Contribution

It establishes a criterion for aritizability of group theories and demonstrates differences for monoids and groupoids, providing new insights into their theoretical properties.

Findings

Group theory is aritizable iff the group is finite.

Monoids and groupoids can have infinite structures with binary theories.

The criterion for aritizability differs between groups and monoids/groupoids.

Abstract

We study applications of a general approach for arities and arizabilities of theories to group and monoid theories. It is proved that a theory of a group $G$ is aritizable if and only if $G$ is finite. It is shown that this criterion does not hold for monoids/groupoids: there is an infinite monoid/groupoid having a binary theory.