Profinite semigroups, Mal'cev products, and identities (Q1922482)

The Mal'cev product \({\mathbf V}\circ{\mathbf W}\) of two pseudovarieties \({\mathbf V}\) and \({\mathbf W}\) of (ordered) semigroups is defined to be the pseudovariety consisting of all finite semigroups \(S\) such that there is a relational morphism \(\varphi: S\to T\) with \(T\in{\mathbf W}\) and, for every idempotent \(e\) in \(T\), \(e\varphi^{-1}\in{\mathbf V}\). Let \(\widehat F_A({\mathbf V})\) denote the free pro-\({\mathbf V}\) semigroup on the (pro)finite set \(A\), \({\mathbf S}\) the pseudovariety of all finite semigroups, and \(\pi:\widehat F_A({\mathbf S})\to\widehat F_A({\mathbf W})\) the canonical projection. Through a compactness argument, the authors show that a finite semigroup \(S\) lies in \({\mathbf V}\circ{\mathbf W}\) if and only if, for every (equivalently, for some) onto continuous homomorphism \(\sigma:\widehat F_A({\mathbf S})\to{\mathbf S}\) and for every idempotent \(e\) in \(\widehat F_A({\mathbf W})\), \(e(\pi^{-1}\sigma)\in{\mathbf V}\). (A similar argument also leads to a characterization of \({\mathbf W}\)-pointlike subsets of \(S\) in terms of the relational morphism \(\sigma^{-1}\pi\).) This allows them to obtain a general theorem describing a basis of (pseudo)identities for \({\mathbf V}\circ{\mathbf W}\) by substituting for all variables in each identity in a given basis for \({\mathbf V}\) elements of \(\widehat F_A({\mathbf S})\) having all the same idempotent image under \(\pi\). As examples of applications, bases of identities are given for specific Mal'cev products and the decidability of \({\mathbf V}\circ{\mathbf J}_1\), \({\mathbf V}\circ{\mathbf N}{\mathbf i}{\mathbf l}\) and \({\mathbf V}\circ{\mathbf J}\) is established when \({\mathbf V}\) is decidable, where \({\mathbf J}_1\), \({\mathbf N}{\mathbf i}{\mathbf l}\) and \({\mathbf J}\) are, respectively, the pseudovarieties of all finite idempotent and commutative semigroups, nilpotent semigroups, and \({\mathcal J}\)-trivial semigroups.