Reprint of: ω-Lyndon words

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