Fixed Points of Endomorphisms and Metrics in preGarside Monoids

Abstract

E. Rodaro and P. V. Silva proved recently 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, the fixed and periodic points submonoids of any endomorphism are also finitely generated left preGarside monoids under some condition, and in the particular case of Artin–Tits monoids, these submonoids are Artin–Tits monoids too. We prove algebraically some inequalities, equivalences, and nonequivalences between three metrics in finitely generated preGarside monoids, and especially in trace monoids and Garside monoids.

Key Words:

• Artin–Tits monoids
• Endomorphisms
• Fixed points
• Garside monoids
• Metrics

2010 Mathematics Subject Classification:

• 20F36