$\omega$-Lyndon words

Let $\text{A}$ be a finite non-empty set and $⪯$ a total order on ${\text{A}}^{\text{nats}}$ verifying the following lexicographic like condition: For each $n\in \text{nats}$ and $u,v\in {\text{A}}^n,$ if $u^{\omega }\prec v^{\omega }$ then $ux\prec vy$ for all $x, y \in \A^\nats.$Aword$x\in \A^\nats$iscalled$\omega$-Lyndonif$x\prec y$foreachpropersuffix$y$of$x.$Afiniteword$w\in \A^+$iscalled$\omega$-Lyndonif$w^\omega \prec v^\omega$foreachpropersuffix$v$of$w.$ In this note we prove that every infinite word may be written uniquely as a non-increasing product of $\omega$-Lyndon words.