Soficity for monoids, semigroups, and general dynamical systems

TL;DR

This paper explores various definitions of soficity for monoids, extends some to semigroups, and proposes a new framework for soficity in general dynamical systems involving semigroup actions.

Contribution

It compares different soficity definitions for monoids, generalizes some to semigroups, and introduces a new concept of soficity for dynamical systems with semigroup actions.

Findings

Not all definitions of soficity for monoids are equivalent.

A semigroup generalization of soficity is proposed.

A new definition of soficity for dynamical systems is introduced.

Abstract

We examine several definitions of soficity for monoids obtained by generalizing various definitions of sofic groups. They are not all equivalent and include the definition recently introduced by Ceccherini-Silberstein and Coornaert. One of these definitions readily generalizes to semigroups (albeit in an arguably unsatisfying way), thus addressing a question asked by Kambites. We conclude by proposing a definition of soficity for a general dynamical system consisting of a semigroup acting by measure-preserving transformations on a probability space.