On Unique Factorization of Non-periodic Words
Brahim Abdenbi

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 on the free group , we show that every non-periodic cyclically reduced word admits a maximal ascent that is uniquely positioned. This provides a cyclic permutation of that decomposes as where is the maximal ascent and is either trivial or a descent. We show that if is not uniquely positioned in , then it must be an internal subword in . Moreover, we show that when is the Magnus ordering, if and only if is monotonic.
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Taxonomy
Topicssemigroups and automata theory · Geometric and Algebraic Topology · Advanced Combinatorial Mathematics
