Free monoids are coherent

TL;DR

This paper proves that free monoids are coherent, meaning their finitely generated subacts of finitely presented acts are also finitely presented, extending the analogy from free rings and answering a longstanding question.

Contribution

It establishes that all free monoids are coherent, filling a gap in the understanding of monoid coherence properties.

Findings

Free monoids are coherent.

Answer to a 1992 open question.

Extension of coherence concepts from rings to monoids.

Abstract

A monoid $S$ is said to be right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. Left coherency is defined dually and $S$ is coherent if it is both right and left coherent. These notions are analogous to those for a ring $R$ (where, of course, $S$-acts are replaced by $R$-modules). Choo, Lam and Luft have shown that free rings are coherent. In this note we prove that, correspondingly, any free monoid is coherent, thus answering a question posed by the first author in 1992.