Some pseudovariety joins involving locally trivial semigroups (Q5953367)

The paper is concerned with the computation of pseudovariety joins of the form \({\mathcal L}{\mathbf I}\vee{\mathbf V}\) where \({\mathcal L}{\mathbf I}\) is the pseudovariety of locally trivial semigroups and \(\mathbf V\) is a sub-pseudovariety of the Mal'cev product \(\mathbf{CR}\circm{\mathbf N}\). It is shown that, if \(\mathbf V\) is defined by the set of pseudoidentities \(\Sigma\) then \({\mathcal L}{\mathbf I}\vee{\mathbf V}\) is defined by all pseudoidentities of the form \(a^\omega xb^\omega=a^\omega yb^\omega\) where \(x=y\in\Sigma\) and \(a,b\) are new symbols. Analogues for \({\mathcal L}{\mathbf I}\) replaced with \(\mathbf K,\mathbf D\) are also obtained. Further it is shown that similar methods apply to pseudovarieties \(\mathbf V\) not necessarily contained in \(\mathbf{CR}\circm{\mathbf N}\), such as \(\mathbf J\), \(\mathbf Z\mathbf E\), \(\mathbf R\) and others.