Markov semigroups, monoids, and groups

TL;DR

This paper explores the properties of Markov and strongly Markov semigroups and monoids, providing examples, characterizations, and examining their behavior under various algebraic constructions.

Contribution

It introduces and compares generalizations of Markov properties to monoids, presents new examples, and investigates their structural and algebraic behaviors.

Findings

All finitely generated commutative semigroups are strongly Markov.

Finitely generated subsemigroups of virtually abelian groups may not be strongly Markov.

A non-Markov monoid can still have a regular language of unique representatives.

Abstract

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.