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 context-freeness of maximal solutions of systems of semi-linear inequalities. 1

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

We study decompositions of a factorial language to catenations of factorial languages and introduce the notion of a canonical decomposition.

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 c-codes, 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 c-code 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.

We prove that given a set X of two nonempty words, a set Y of nonempty words commutes with X if and only if either Y is a union of powers of X or X;Y ` t + for some primitive word t. We also show that the same holds for certain special types of codes, but does not hold in general for sets of cardinality at least four. 1 Introduction This note deals with a special case of the following general problem. Given a subset of a free semigroup describe, if possible, all subsets which commute with it. We solve the problem when the given subset has exactly two elements. A simple sufficient condition under which two arbitrary elements of an associative algebra commute is when the two elements belong to the subalgebra generated by a third element. In favorable situations this condition is also necessary. This is precisely what happens for polynomials and series of noncommuting variables over a field with Bergman's Theorem [5], for words in free monoids with the Defect Theorem, elements in ...

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

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.

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.

We consider equations on the monoid of factorial languages on the binary alphabet. We use the notion of a canonical decomposition of a factorial language and previous results by Avgustinovich and the author to solve several simple equations on binary factorial languages including X n = Y n, the commutation equation XY = Y X and the conjugacy equation XZ = ZY. At the end of the paper we discuss the difficulties hindering to reduce equations on factorial languages to equations on words and to enlarge the alphabet considered.

We consider equations on the monoid of factorial languages on the binary alphabet. We use the notion of a canonical decomposition of a factorial language and previous results by Avgustinovich and the author to solve several simple equations on binary factorial languages including X n = Y n, the commutation equation XY = Y X and the conjugacy equation XZ = ZY. At the end of the paper we discuss the difficulties hindering to reduce equations on factorial languages to equations on words and to enlarge the alphabet considered.