On periodic points of free inverse monoid endomorphisms

TL;DR

This paper investigates the structure of periodic and fixed points in free inverse monoid endomorphisms, showing the former is finitely generated and the latter can be context-sensitive but not necessarily context-free.

Contribution

It proves the finite generation of periodic point submonoids and characterizes the language complexity of fixed point submonoids in free inverse monoids.

Findings

Periodic point submonoid is always finitely generated

Fixed point submonoid can be represented by a context-sensitive language

Fixed point submonoid cannot always be represented by a context-free language

Abstract

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.