Twisted associativity of the cyclically reduced product of words, part 1

TL;DR

This paper investigates the cyclically reduced product of words, demonstrating a special case where a generalized associative property holds, contributing to the understanding of its algebraic structure.

Contribution

It proves a special case of a generalized associative property for the cyclically reduced product of words, extending previous structural insights.

Findings

A generalized associative property holds in a specific case.

The structure of cyclically reduced words is further understood.

Foundation laid for proving full associativity in future work.

Abstract

The cyclically reduced product of two words $u, v$, denoted $u * v$, is the cyclically reduced form of the concatenation of $u$ by $v$. This product is not associative. Recently S. V. Ivanov has proved that the Andrews-Curtis conjecture can be restated in terms of the cyclically reduced product and cyclic permutations instead of the reduced product and conjugations. In a previous paper we have started a thorough study of $*$ and of the structure of the set of cyclically reduced words $\hat{\mathcal{F}}(X)$ equipped with $*$. In particular we have found that a certain number of properties of the free group equipped with the reduced product can be generalized to $(\hat{\mathcal{F}}(X), *)$. In this paper we continue this study by proving that a generalized version of the associative property holds for $*$ in a special case. In a following paper we will prove that a more general version of the associative property holds for any case.