The Commutation with Codes and Ternary Sets of Words

We prove several results on the commutation of languages. First, we prove that the largest set commuting with a given code X, i.e., its centralizer e(X), is always �(X)*, where �(X) is the primitive root of X. Using this result, we characterize the commutation with codes similarly as for words, polynomials, and formal power series: a language commutes with X if and only if it is a union of powers of �(X). This solves a conjecture of Ratoandromanana, 1989, and also gives an affirmative answer to a special case of an intriguing problem raised by Conway in 1971. Second, we prove that for any nonperiodic ternary set of words F � �+, e(F) = F*, and moreover, a language commutes with F if and only if it is a union of powers of F, results previously known only for ternary codes. A boundary point is thus established, as these results do not hold for all languages with at least four words.