Takahasi semigroups

TL;DR

This paper generalizes Takahasi's theorem from free groups to various classes of semigroups, demonstrating properties of subsemigroups and periodic points using automata and group arguments.

Contribution

It extends Takahasi's theorem to semigroups like completely simple and Clifford semigroups, showing boundedness of periodic orbits and finite generation of periodic points.

Findings

Subsemigroups of periodic points are finitely generated

Periodic orbits are bounded for arbitrary endomorphisms

Results apply to classes like completely simple and Clifford semigroups

Abstract

Takahasi's theorem on chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations. In addition to the background on semigroups, proofs involve arguments over groups and finite automata.