Characterization of group radicals with an application to Mal'cev products.
The authors study Mal'cev products of the form \(\mathbf{LH}\circm\mathbf V\) where \(\mathbf{LH}\) is the pseudovariety of all finite semigroups whose local submonoids belong to a given pseudovariety \(\mathbf H\) of groups and \(\mathbf V\) is a pseudovariety of semigroups. In an earlier paper [Trans.\ Am.\ Math.\ Soc. 361, No. 3, 1429-1461 (2009; Zbl 1185.20058)], they have described such products in the case when \(\mathbf H\) either is trivial or coincides with the pseudovariety of all finite \(p\)-groups for some prime \(p\). Here the authors extend the description to the case when \(\mathbf H\) is a group pseudovariety being simultaneously a Fitting class (Theorem~5.2). They also show how to obtain a basis of pseudoidentities for \(\mathbf{LH}\circm\mathbf V\) from a basis of pseudoidentities for \(\mathbf V\) and a certain profinite characterization of the \(\mathbf H\)-radical (Theorem~6.1).
scientific article; zbMATH DE number 5882254