Free profinite semigroups over semidirect products
Free profinite semigroups and monoids over semidirect products are studied in the paper. A general description of these free objects over profinite spaces of generators is given in section 2. In case of semidirect products of the form \(J_1*V\) (where \(J_1\) is the pseudovariety of finite semilattice monoids) the description is interpreted in terms of profinite graphs (Section 3); also analogous results for the two-sided semidirect products are given in Section 5. Using this the authors make a complete study of the pseudovariety \(J_1*\text{Nil}\) (\(\text{Nil}\) is the pseudovariety of the finite nilpotent semigroups), i.e. give pseudoidentities defining \(J_1*\text{Nil}\), give a description of the corresponding free profinite semigroups, solve the word problem for them, give a description of the variety of rational languages associated with \(J_1*\text{Nil}\) (Section 4). In section 6 the authors construct the free profinite inverse monoid on a profinite alphabet \(A\) and show that the free profinite monoid over \(J_1*G\) (\(G\) is the pseudovariety of all finite groups) on \(A\) can be identified with one of its closed submonoids. Both these profinite monoids are described in terms of the Cayley graphs of the free profinite group.
free objects over profinite spaces of generators
scientific article; zbMATH DE number 862274