Semidirect products of ordered semigroups

Jean-Eric Pin (1) , Pascal Weil (2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

Semidirect product is a powerful tool for studying finite semigroups, and it has been used in the literature to give structure theorems and classification theorems, especially in the context of the lattice of varieties of finite semigroups. This study in turn has deep connections with the classification of recognizable languages. Our aim in this paper is to develop a body of results on the semidirect product of (finite) ordered semigroups, that can be used like the more classical results on finite unordered structures. A pioneering --- and inspiring --- work in this direction is briefly sketched at the end of the paper by Straubing and Thérien (1985), but it long looked like an isolated attempt. The main results of the paper can be considered as the ordered counterparts of certain classical results on semigroups. We cover in particular two decomposition theorems which were much discussed in the literature throughout the 1980s, until they were finally proved by brilliant results of Ash. The first of these results deals with the variety generated by inverse monoids, and the second one with the variety of so-called block-groups. Applications of the results of the present paper to language theory are deferred to the article The wreath product principle for ordered semigroups.

Mots clés: Semigroup; semidirect product; ordered semigroup

Domaines: Autre [cs.OH]; Théorie des groupes [math.GR]

Jean-Eric Pin, Pascal Weil. Semidirect products of ordered semigroups. Communications in Algebra, 2002, 30, pp.149-169. ⟨hal-00112613⟩