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24
The power of commuting with finite sets of words
 In Proc. STACS’05, Springer LNCS 3404
, 2005
"... We construct a finite language L such that the largest language commuting with L is not recursively enumerable. This gives a negative answer to the question raised by Conway in 1971 and also strongly disproves Conway’s conjecture on contextfreeness of maximal solutions of systems of semilinear ine ..."
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We construct a finite language L such that the largest language commuting with L is not recursively enumerable. This gives a negative answer to the question raised by Conway in 1971 and also strongly disproves Conway’s conjecture on contextfreeness of maximal solutions of systems of semilinear inequalities. 1
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
Challenges of Commutation  An Advertisement
, 2001
"... We consider a few problems connected to the commutation of languages, in particular finite ones. The goal is to emphasize the challenging nature of such simply formulated problems. In doing so we give a survey of results achieved during the last few years, restate several open problems and illustr ..."
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Cited by 4 (3 self)
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We consider a few problems connected to the commutation of languages, in particular finite ones. The goal is to emphasize the challenging nature of such simply formulated problems. In doing so we give a survey of results achieved during the last few years, restate several open problems and illustrate some approaches to attack such problems by two simple constructions.
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.
Simple language equations
 Bull. Eur. Assoc. Theor. Comput. Sci. EATCS
, 2005
"... Abstract. We survey results, both positive and negative, on regularity of maximal solutions of systems of implicit language equations and inequalities. These results concern inequalities with constant righthand sides, onesided linear inequalities, inequalities with restrictions on constants, and c ..."
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Abstract. We survey results, both positive and negative, on regularity of maximal solutions of systems of implicit language equations and inequalities. These results concern inequalities with constant righthand sides, onesided linear inequalities, inequalities with restrictions on constants, and commutation equations and inequalities. In addition, we present some of these results in a generalized form in order to underline common principles. 1.
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, is ..."
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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, is
AUTOMORPHISMS OF THE SEMIGROUP OF ENDOMORPHISMS OF FREE ASSOCIATIVE ALGEBRAS
, 2005
"... Abstract. Let A = A(x1,...,xn) be a free associative algebra in the variety of associative algebras A freely generated over K by a set X = {x1,...,xn}, End A be the semigroup of endomorphisms of A, and Aut EndA be the group of automorphisms of the semigroup EndA. We investigate the structure of the ..."
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Abstract. Let A = A(x1,...,xn) be a free associative algebra in the variety of associative algebras A freely generated over K by a set X = {x1,...,xn}, End A be the semigroup of endomorphisms of A, and Aut EndA be the group of automorphisms of the semigroup EndA. We investigate the structure of the groups Aut EndA and Aut A ◦ , where A ◦ is the category of finitely generated free algebras from A. We prove that the group Aut EndA is generated by semiinner and mirror automorphisms of EndF and the group Aut A ◦ is generated by semiinner and mirror automorphisms of the category A ◦. This result solves an open Problem formulated in [14]. 1.
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
TEST ELEMENTS, RETRACTS AND AUTOMORPHIC ORBITS
, 807
"... Abstract. Let A2 be a free associative or polynomial algebra of rank two over a field K of characteristic zero. Based on the degree estimate of MakarLimanov and J.T.Yu, we prove: 1) An element p ∈ A2 is a test element if p does not belong to any proper retract of A2; 2) Every endomorphism preservi ..."
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Abstract. Let A2 be a free associative or polynomial algebra of rank two over a field K of characteristic zero. Based on the degree estimate of MakarLimanov and J.T.Yu, we prove: 1) An element p ∈ A2 is a test element if p does not belong to any proper retract of A2; 2) Every endomorphism preserving the automorphic orbit of a nonconstant element of A2 is an automorphism. 1. Introduction and