Pointlike reducibility of pseudovarieties of the form \(\mathbf V*\mathbf D\). (Q2799118)

Given elements \(s_1,\ldots,s_n\) from a finite semigroup \(S\) and a continuous homomorphism \(\varphi\) from a free profinite semigroup \(F\) onto \(S\), the pointlike problem with respect to a pseudovariety of semigroups \(\mathbf V\) consists in determining whether there are elements \(w_1,\ldots,w_n\) from \(F\) such that \(\varphi(w_i)=s_i\) (\(i=1,\ldots,n\)) and the pseudoidentities \(w_1=\cdots=w_n\) are valid in \(\mathbf V\).NEWLINENEWLINE Following seminal work of \textit{C. J. Ash} [Int. J. Algebra Comput. 1, No. 1, 127-146 (1991; Zbl 0722.20039)] for the pseudovariety of all finite groups, a method introduced by \textit{B. Steinberg} and the reviewer [Proc. Lond. Math. Soc., III. Ser. 80, No. 1, 50-74 (2000; Zbl 1027.20033)] for solving such problems consists in finding a suitably computable restricted signature \(\sigma\) which contains enough elements to witness all positive cases of pointlike problem for \(\mathbf V\), a property that is called \textit{pointlike \(\sigma\)-reducibility}.NEWLINENEWLINE The main result of the present paper states that, if \(\mathbf V\) is pointlike \(\sigma\)-reducible, then so is the semidirect product \(\mathbf V*\mathbf D\), where \(\mathbf D\) stands for the pseudovariety of all finite semigroups in which idempotents are right zeros.