ω-Lyndon words

Let A be a finite non-empty set and ⪯ a total order on AN verifying the following lexicographic like condition: For each n∈N and u,v∈An, if uω≺vω then ux≺vy for all x,y∈AN. A word x∈AN is called ω-Lyndon if x≺y for each proper suffix y of x. A finite word w∈A+ is called ω-Lyndon if wω≺vω 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.