Almost overlap-free words and the word problem for the free Burnside semigroup satisfying x^2=x^3

TL;DR

This paper introduces a linear-time algorithm for solving the word problem in a specific Burnside semigroup by using almost overlap-free words as canonical representatives, generalizing previous methods.

Contribution

It demonstrates that almost overlap-free words can serve as canonical forms, enabling an efficient partial solution to the word problem for the semigroup.

Findings

Linear-time algorithm for the word problem

Almost overlap-free words as canonical representatives

Efficient partial solution to the word problem

Abstract

In this paper we investigate the word problem of the free Burnside semigroup satisfying x^2=x^3 and having two generators. Elements of this semigroup are classes of equivalent words. A natural way to solve the word problem is to select a unique "canonical" representative for each equivalence class. We prove that overlap-free words and so-called almost overlap-free words (this notion is some generalization of the notion of overlap-free words) can serve as canonical representatives for corresponding equivalence classes. We show that such a word in a given class, if any, can be efficiently found. As a result, we construct a linear-time algorithm that partially solves the word problem for the semigroup under consideration.