$ω$-Lyndon words

Abstract

Let $\A$ be a finite non-empty set and $\preceq $ a total order on $\A^\nats$ verifying the following lexicographic like condition: For each $n\in \nats$ and $u, v\in \A^n,$ if $u^ω\prec v^ω$ then $ux\prec vy$ for all $x, y \in \A^\nats.$ A word $x\in \A^\nats$ is called $ω$-Lyndon if $x\prec y$ for each proper suffix $y$ of $x.$ A finite word $w\in \A^+$ is called $ω$-Lyndon if $w^ω\prec v^ω$ for each proper suffix $v$ of $w.$ In this note we prove that every infinite word may be written uniquely as a non-increasing product of $ω$-Lyndon words.