On sofic monoids

TL;DR

This paper explores the concept of soficity in monoids, establishing which classes are sofic and demonstrating that some finitely presented amenable inverse monoids are non-sofic, unlike groups.

Contribution

It extends the notion of soficity from groups to monoids, identifying classes that are sofic and providing the first example of a non-sofic finitely presented amenable inverse monoid.

Findings

All finite, commutative, free, cancellative one-sided amenable, matrix, and identity-augmented monoids are sofic.

The bicyclic monoid is not sofic, providing a counterexample.

Some finitely presented amenable inverse monoids are non-sofic.

Abstract

We investigate the notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup without identity element are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.