Fixed points of endomorphisms and relations between metrics in preGarside monoids

TL;DR

This paper extends the understanding of fixed and periodic points in preGarside monoids, showing they are finitely generated under certain conditions and exploring metric relations within these algebraic structures.

Contribution

It demonstrates that fixed and periodic points submonoids of endomorphisms are finitely generated preGarside monoids, including Artin monoids, and analyzes metric inequalities in these contexts.

Findings

Fixed and periodic points submonoids are finitely generated preGarside monoids.

In Artin monoids, these submonoids are also Artin monoids.

Established inequalities and relations between three metrics in preGarside monoids.

Abstract

Rodaro and Silva proved that the fixed points submonoid and the periodic points submonoid of a trace monoid endomorphism are always finitely generated. We show that for finitely generated left preGarside monoids, that includs finitely generated preGarside monoids, Garside monoids and Artin monoids, the fixed and periodic points submonoids of any endomorphism are also finitely generated left preGarside monoids under some condition, and in the case of Artin monoids, these submonoids are always Artin monoids too. We also prove algebraically some inequalities, equivalences and non-equivalences between three metrics in finitely generated preGarside monoids, and especially in trace monoids and Garside monoids.