Free Profinite ℛ-Trivial Monoids

Abstract

This article is concerned with the structure of semigroups of implicit operations on R, the pseudovariety of all ℛ-trivial semigroups. We give two complementary descriptions for these semigroups: first by means of labeled ordinals, and second by means of labeled infinite trees of finite depth. In the first representation, the product of implicit operations has a simple characterization. In the second one, the topology on implicit operations is best visible. A similar study is conducted for the implicit operations on DRG, the non-aperiodic analogue of R. This leads to a description of the corresponding variety of recognizable languages and to the computation of certain joins of pseudovarieties.