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 C(X), is always ρ(X)., where ρ(X) is the primitive rootof 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 ρ(X). This solves a conjecture of Ratoandromanana, 1989, and also gives an affermative 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 ⊆∑+,C(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.
Topics:Regular Languages Combinatories on Words
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