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Conway's Problem and the commutation of languages
 Bulletin of EATCS
, 2001
"... We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The se ..."
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Cited by 8 (5 self)
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We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The second problem, proposed by Ratoandromanana in 1989, is asking for a characterization of those languages commuting with a given code – the conjecture is that the commutation with codes may be characterized as in free monoids. We present here simple proofs for the known results on these two problems. 1
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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Cited by 4 (3 self)
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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.
On the Centralizer of a Finite Set
 in Proc. of ICALP 2000, LNCS 1853
, 2000
"... We prove two results on commutation of languages. First, we show that the maximal language commuting with a three element language, i.e. its centralizer, ..."
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Cited by 3 (2 self)
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We prove two results on commutation of languages. First, we show that the maximal language commuting with a three element language, i.e. its centralizer,
Commutation with codes
 Theor. Comput. Sci
, 2005
"... The centralizer of a set of words X is the largest set of words C(X) commuting with X: XC(X) = C(X)X. It has been a long standing open question due to Conway, 1971, whether the centralizer of any rational set is rational. While the answer turned out to be negative in general, see Kunc 2004, we prov ..."
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Cited by 2 (1 self)
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The centralizer of a set of words X is the largest set of words C(X) commuting with X: XC(X) = C(X)X. It has been a long standing open question due to Conway, 1971, whether the centralizer of any rational set is rational. While the answer turned out to be negative in general, see Kunc 2004, we prove here that the situation is different for codes: the centralizer of any rational code is rational and if the code is finite, then the centralizer is finitely generated. This result has been previously proved only for binary and ternary sets of words in a series of papers by the authors and for prefix codes in an ingenious paper by Ratoandromanana 1989 – many of the techniques we use in this paper follow her ideas. We also give in this paper an elementary proof for the prefix case. Key words: Codes, Commutation, Centralizer, Conway’s problem, Prefix codes. 1
On Commutation and Primitive Roots of Codes
, 2001
"... As is well known, two words commute if and only if they are powers of a same word; two formal power series with coecients in a eld commute if and only if they are combinations of a third series. On the other hand, almost nothing is known about the commutation of sets of words. We give in this paper ..."
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Cited by 1 (0 self)
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As is well known, two words commute if and only if they are powers of a same word; two formal power series with coecients in a eld commute if and only if they are combinations of a third series. On the other hand, almost nothing is known about the commutation of sets of words. We give in this paper a similar characterization for the commutation with an omega code, and for the commutation of two codes. Using these results, we solve a conjecture of Ratoandromanana stating that any code has a unique primitive root. We also prove that the centralizer of a regular omega code, i.e., the maximal set commuting with it, is regular, answering positively in a special case of the problem proposed by Conway more than 30 years ago.
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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Cited by 1 (1 self)
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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.
Contents
"... Contents 9 Centralizers of noncommutative series and polynomials 1 9.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9.1 Cohn's centralizer theorem . . . . . . . . . . . . . . . . . . . . . 2 9.2 Euclidean division and principal right ideals . . . . . . . . . . . 4 9.3 In ..."
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Contents 9 Centralizers of noncommutative series and polynomials 1 9.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9.1 Cohn's centralizer theorem . . . . . . . . . . . . . . . . . . . . . 2 9.2 Euclidean division and principal right ideals . . . . . . . . . . . 4 9.3 Integral closure of the centralizer . . . . . . . . . . . . . . . . . . 5 9.4 Homomorphisms into k[t] . . . . . . . . . . . . . . . . . . . . . . 7 9.5 Bergman's centralizer theorem . . . . . . . . . . . . . . . . . . . 8 9.6 Free subalgebras and the defect theorem . . . . . . . . . . . . . 9 9.7 Appendix: some commutative algebra . . . . . . . . . . . . . . . 11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Bibliography 17 9.0. Introduction It is a wellknown and not too difficult result of combinatorics on words that if two words commute under the concatenation product, then they are both power of the same word: they have a common root. This fact is essentially