On Unique Factorization of Non-periodic Words

TL;DR

This paper investigates the unique factorization properties of non-periodic words in free groups under a bi-order, revealing conditions for unique maximal ascent and descent decompositions, especially in the context of the Magnus ordering.

Contribution

It introduces a novel factorization framework for non-periodic words in free groups based on maximal ascent and descent, with uniqueness conditions under bi-orders.

Findings

Every non-periodic cyclically reduced word admits a unique maximal ascent.

If the descent is not uniquely positioned, it must be an internal subword in the ascent.

Under Magnus ordering, the descent is trivial if and only if the word is monotonic.

Abstract

Given a bi-order $\succ$ on the free group $\mathcal{F}$, we show that every non-periodic cyclically reduced word $W\in \mathcal{F}$ admits a maximal ascent that is uniquely positioned. This provides a cyclic permutation of $W'$ that decomposes as $W'=AD$ where $A$ is the maximal ascent and $D$ is either trivial or a descent. We show that if $D$ is not uniquely positioned in $W$, then it must be an internal subword in $A$. Moreover, we show that when $\succ$ is the Magnus ordering, $D=1_\mathcal{F}$ if and only if $W$ is monotonic.