Carrying on the work of Arnold, Pécuchet and Perrin, Wilke has obtained a counterpart of Eilenberg’s variety theorem for finite and infinite words. In this paper, we extend this theory for classes of languages that are closed under union and intersection, but not necessarily under complement. As an example, we give a purely algebraic characterization of various classes of recognizable sets defi...

Let 5 be a commutative semigroup. For a, b £5, we say that a divides b (or b is a multiple of a), and write a\b, if either a = b or ax = b for some x£5. We say that a properly divides b if a\b and b does not divide a. We call 5 archimedean if for all a, b £5, a divides some power of b. It is known that every commutative semigroup is uniquely expressible as a semilattice of archimedean semigroup...

By an ordered commutative semigroup of the first kind (abbreviated "o.c.s.I") we mean a system 5(o, <) consisting of a set 5 endowed with a binary operation o and a binary relation < such that the following axioms are satisfied. I. S is a commutative semigroup with respect to o, i.e. the associative law, a o (b o c) = (a o b)o c, and the commutative law, a o b = b o a, hold (a, b, c in S). II. ...

Straubing’s wreath product principle provides a description of the languages recognized by the wreath product of two monoids. A similar principle for ordered semigroups is given in this paper. Applications to language theory extend standard results of the theory of varieties to positive varieties. They include a characterization of positive locally testable languages and syntactic descriptions ...

In this note, we show in the line of [3] that an ordered Γ-semigroup is left and right simple if and only if it does not contain proper bi-ideals. Mathematics Subject Classification: 06F99