Do Long Van

Codes and length-increasing transitive binary relations

Classes of codes defined by binary relations are considered. It turns out that many classes of codes can be defined by length-increasing transitive binary relations. By introducing a general embedding schema we show that the embedding problem can be solved in a unified way for many classes of codes defined in such a way. Several among these classes of codes can be characterized by means of variants of Parikh vectors. This is very useful in constructing many-word concrete codes, maximal codes in corresponding classes of codes. Also, this allows to establish procedures to generate all maximall codes as well as algorithms to embed a code in a maximal one in some classes of codes.

Characterizations of rational Ω-languages by means of right congruences

Abstract By means of right congruences, we characterize in a unified way different classical classes of rational ω-languages. A new congruence associated with an ω-automaton is introduced named cycle congruence. The family of rational ω-languages which have, by morphism, a unique minimal recognizing ω-automaton is characterized. It appears that for such a minimal ω-automaton, the cycle congruence coincides with the syntactic congruence of the recognized ω-language. We prove that the other rational ω-languages have an infinite number of minimal automaton.

Stability for the zigzag submonoids

Abstract A notion of stability for the zigzag submonoids is introduced and called Z-stability. We prove that for the zigzag submonoids the Z-stability and the Z-freeness are equivalent, like the stability and the freeness for the ordinary submonoids. The class of Z-free zigzag submonoids is, however, not closed under intersection. We prove also that the property of being Z-stable is decidable for the regular zigzag submonoids.

Ensembles code-compatibles et une généralisation du théorème de Sardinas-Patterson

One denotes by A∞ the monoid of all finite and infinite words over an alphabet A. The code-compatibility of a pair (X, Y) of subsets of A∞ is here defined. A necessary and sufficient condition, which is a generalization of the Sardinas-Patterson theorem, for (X, Y) to be code-compatible is established. It is shown that the class of the submonoids of A∞ generated by an infinitary code is closed by intersection, and also that, for any two infinitary codes X and Y, X∗ ∩ Y∗ = (X ∩ Y)∗iff (X, Y) is code-compatible.

Varieties of finite monoids and Bu¨chi-McNaughton theorem

Abstract Perrin (1982) has proved that a part of Buchi-McNaughton theorem in the form formulated by Eilenberg (1974) can be generalized to a large class of varieties of recognizable languages. We show here that Perrins assertion can be extended to the whole theorem as well as to the partial improvements of it given by Arnold (1983), Beauquier and Perrin (1984). To do that, suitable classes of automata associated with a variety of finite monoids are introduced and examined.