Weil, Pascal; Almeida, Jorge Free profinite \(\mathcal R\)-trivial monoids. (English) Zbl 0892.20035 Int. J. Algebra Comput. 7, No. 5, 625-671 (1997). The structure of the free profinite \(\mathcal R\)-trivial monoids is elucidated in this paper. If \(\mathbf V\) is a pseudovariety of finite monoids, the free pro-\(\mathbf V\) monoid over an alphabet \(A\) is the projective limit of the \(A\)-generated elements of \(\mathbf V\). The algebraic and topological structure of the free pro-\(\mathbf V\) monoids illuminates several questions on the class of rational languages recognized by monoids in \(\mathbf V\), and on the membership problem for pseudovarieties built from \(\mathbf V\) using such operations as joins, semidirect products and Mal’cev products. However, the structure of the free pro-\(\mathbf V\) monoids is known for relatively few values of \(\mathbf V\).In this paper, two equivalent representations of the free pro-\(\mathbf R\) monoids are given, where \(\mathbf R\) is the pseudovariety of \(\mathcal R\)-trivial monoids. The first representation is in terms of labeled ordinals, or transfinite words, and it renders more adequately by algebraic properties of these monoids. The second one, in terms of labeled trees is more adapted to render their topological properties. The method is then extended to describe the structure of the free pro-\(\mathbf{DRH}\) monoids, where \(\mathbf H\) is a pseudovariety of groups and \(\mathbf{DRH}\) is the pseudovariety of all semigroups in which each regular \(\mathcal R\)-class is a group in \(\mathbf H\). These results are applied to the determination of the languages recognized by monoids in \(\mathbf{DRH}\) and to the computation of the join of \(\mathbf{DRH}\) and its right-left dual \(\mathbf{DLH}\). Reviewer: P.Weil (Paris) Cited in 2 ReviewsCited in 12 Documents MSC: 20M05 Free semigroups, generators and relations, word problems 20M07 Varieties and pseudovarieties of semigroups 20M35 Semigroups in automata theory, linguistics, etc. 68Q70 Algebraic theory of languages and automata 08A70 Applications of universal algebra in computer science 08C15 Quasivarieties Keywords:pseudovarieties of semigroups; free profinite \(\mathcal R\)-trivial monoids; pseudovarieties of finite monoids; rational languages; membership problem; Mal’cev products; labeled ordinals; labeled trees; pseudovarieties of groups PDFBibTeX XMLCite \textit{P. Weil} and \textit{J. Almeida}, Int. J. Algebra Comput. 7, No. 5, 625--671 (1997; Zbl 0892.20035) Full Text: DOI