On Unique Factorization of Non-periodic Words

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 $W^{\mathrm{\prime }}$ that decomposes as $W^{\mathrm{\prime }}=AD$ 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.