Square-free reducts of words

TL;DR

This paper studies the process of reducing words by removing squares to reach square-free words, revealing that ternary words can have many reducts and classifying words into finitely many related classes.

Contribution

It proves the existence of ternary words with arbitrarily many reducts and characterizes the structure of related word classes under square reductions.

Findings

Binary words have exactly one reduct.

Ternary words can have arbitrarily many reducts.

The number of reducts for ternary words grows exponentially.

Abstract

A \emph{square} is a finite non-empty word consisting of two identical adjacent blocks. A word is \emph{square-free} if it does not contain a square as a factor. In any finite word one may delete the repeated block of a square, obtaining thereby a shorter word. By repeating this process, a square-free word is eventually reached, which we call a \emph{reduct} of the original word. How many different reducts a single word may have? It is not hard to prove that any binary word has exactly one reduct. We prove that there exist ternary words with arbitrarily many reducts. Moreover, the function counting the maximum number of reducts a ternary word of length $n$ may have grows exponentially. We also prove that over four letters, there exist words with any given number of reducts, which does not seem to be the case for ternary words. Finally, we demonstrate that the set of all finite ternary words splits into finitely many classes of \emph{related} words (one may get from one word to the other by a sequence of square reductions and factor duplications). A few open questions are posed concerning some structures on words defined with the use of the square reduction.