Extensions by local groups (Q1910267)

Given pseudovarieties \(\mathbf V\) and \(\mathbf W\) of semigroups, let \({\mathbf V}^{-1} {\mathbf W}\) be the class of all finite semigroups \(S\) for which there is a relational morphism \(\tau : S \to T\) with \(T \in {\mathbf W}\) such that, for every subsemigroup \(U\) of \(T\) which lies in \(\mathbf V\), \(\tau^{-1} U\) also belongs to \(\mathbf V\). For some pseudovarieties \(\mathbf V\), which are called good, it suffices to consider trivial subsemigroups \(U\) in the above definition. So, for a good pseudovariety \(\mathbf V\), \({\mathbf V}^{-1} {\mathbf W}\) is the Mal'cev product of \(\mathbf V\) with \(\mathbf W\). Another possible simplification is obtained when instead of arbitrary relational morphisms, ordinary homomorphisms \(\tau\) suffice in the definition of \({\mathbf V}^{-1} {\mathbf W}\), in which case if both \(\mathbf V\) and \(\mathbf W\) have decidable membership problems, then so does \({\mathbf V}^{-1} {\mathbf W}\). The main theorem in the paper under review shows that such a simplification holds when \(\mathbf V\) is the pseudovariety of all finite local groups (which is good) and also gives a basis of pseudoidentities for \({\mathbf V}^{-1} {\mathbf W}\) whenever such a basis is known for \(\mathbf W\). A generalization of this result is also discussed.