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Commutation Problems on Sets of Words and Formal Power Series
, 2002
"... We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generaliza ..."
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We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language  the largest set commuting with a language , of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago  asking whether or not the centralizer of any rational language is rational  in the case of periodic, binary, and ternary sets of words, as well as for rational ccodes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an ccode X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.
The Commutation With Codes and Ternary Sets of Words
, 2002
"... 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 (X) , where (X) is the primitive root of X . Using this result, we characterize the commutation with codes similarly as for words, p ..."
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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 (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 armative 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 , 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 languages with at least four words.