Commutation with Ternary Sets of Words

We prove that for any nonperiodic set of words F \subseteq Σ+ with at most three elements, the centralizer of F, i.e., the largest set commuting with F, is F*. Moreover, any set X commuting with F is of the form X = FI, for some I \subseteq ℕ. A boundary point is thus established, as these results do not hold for all languages with at least four words. This solves a conjecture of Karhumaki and Petre, and provides positive answers to special cases of some intriguing questions on commutation of languages, raised by Ratoandromanana and Conway.