Acyclic, connected and tree sets

TL;DR

This paper explores special classes of word sets characterized by properties of their extension graphs, establishing connections with free groups and subgroup bases, with implications for combinatorics on words.

Contribution

It characterizes acyclic and tree sets of words via extension graph properties and links first return words to free group bases in uniformly recurrent tree sets.

Findings

In uniformly recurrent tree sets, first return words form free group bases.

A set is acyclic if and only if all included bifix codes are subgroup bases.

Provides a characterization of acyclic, connected, and tree sets through extension graph properties.

Abstract

Given a set $F$ of words, one associates to each word $w$ in $F$ an undirected graph, called its extension graph, and which describes the possible extensions of $w$ on the left and on the right. We investigate the family of sets of words defined by the property of the extension graph of each word in the set to be acyclic or connected or a tree. We prove that in a uniformly recurrent tree set, the sets of first return words are bases of the free group on the alphabet. Concerning acyclic sets, we prove as a main result that a set $F$ is acyclic if and only if any bifix code included in $F$ is a basis of the subgroup that it generates.