Free profinite \(\mathcal R\)-trivial, locally idempotent and locally commutative semigroups
For each of the Green relations \({\mathcal T}\in\{\mathcal{R,L,J}\}\), let \(\mathbf T\) denote the pseudovariety of all finite \(\mathcal T\)-trivial semigroups. Let also \(\mathbf{DA}\) denote the pseudovariety of all finite semigroups in which regular elements are idempotents and \(\mathbf{LJ}_1\) the pseudovariety of all finite semigroups which are locally semilattices. The central results in the paper describe the finitely generated free profinite semigroups over the intersection \({\mathbf R}\cap\mathbf{LJ}_1\). For these semigroups, a normal factorization in terms of generators and idempotents is given, idempotents being characterized by means of their content and their restriction to semigroups in which idempotents are left zeros. These results yield corresponding results for free profinite semigroups over \({\mathbf J}\cap\mathbf{LJ}_1\), which reproves results of \textit{C.~Selmi} [Theor. Comput. Sci. 161, No. 1-2, 157-190 (1996; Zbl 0872.68092)], and \(\mathbf{DA}\cap\mathbf{LJ}_1\), and allow the author to prove that the latter intersection is the join of \({\mathbf R}\cap\mathbf{LJ}_1\) with \({\mathbf L}\cap\mathbf{LJ}_1\). Further applications are obtained by interpreting these results and their proofs in terms of varieties of rational languages. (Also submitted to MR).
scientific article; zbMATH DE number 1309716
scientific article