Unary operations on pseudo-varieties of semigroups (Q1910270)

An \(n\)-ary implicit operation in a semigroup pseudovariety \(\mathcal V\) is a mapping \(\pi\) associating to each \(S\in{\mathcal V}\) an \(n\)-ary function \(\pi_S:S^n\to S\) on \(S\) such that \(\pi\) preserves every homomorphism \(f:S\to T\) between semigroups \(S,T\in{\mathcal V}\) in the sense that \(f(\pi_S(s_1,\dots,s_n))=\pi_T(f(s_1),\dots,f(s_n))\) for all \(s_1,\dots,s_n\in S\). The set \(\overline{F}_n{\mathcal V}\) of all \(n\)-ary implicit operations in \(\mathcal V\) is a topological semigroup with respect to pointwise multiplication and the least topology such that all homomorphisms from \(\overline{F}_n{\mathcal V}\) into (discrete) semigroups of \(\mathcal V\) are continuous. It is well known that, for semigroup pseudovarieties, semigroups of implicit operations are substitutes for free objects. The paper under review is aimed at a description of the semigroup \(\overline{F}_1 {\mathcal V}\) of unary implicit operations in \(\mathcal V\). For the pseudovariety \(\mathcal S\) of all finite semigroups, the unary implicit operations were described by \textit{J. Almeida} and \textit{A. Azevedo} [in Proc. Int. Conf., Chico/Calif. 1986, 1-11 (1987; Zbl 0623.20041)]. They proved that \(\overline{F}_1{\mathcal S}\) equals the union of the free monogenic subsemigroup \(F_1{\mathcal S}\) with the group ideal \(I_1{\mathcal S}\) and characterized the implicit operations in \(I_1{\mathcal S}\) via infinite sequences of functions from the set of all primes \(\mathbb{P}\) into the set of all non-negative integers. The author shows (Theorem 1) that the group \(I_1{\mathcal S}\) is isomorphic (as a topological group) to the direct product \(\widehat{Z}\) of the topological groups of \(p\)-adic integers where \(p\) runs over \(\mathbb{P}\). He also describes how to multiply the elements of \(I_1{\mathcal S}\) by those from \(F_1{\mathcal S}\) (Proposition 3). Then, for an arbitrary semigroup pseudovariety \(\mathcal V\), he describes the kernel of the natural homomorphism of \(\overline{F}_1{\mathcal S}\) onto \(\overline{F}_1{\mathcal V}\) (Theorem 2) thus clarifying the structure of the latter semigroup. As a corollary, he classifies semigroup pseudovarieties \(\mathcal V\) such that \(\overline{F}_1{\mathcal V}\) is finite or is countable or has the cardinality of the continuum (Proposition 4). Reviewer's remarks: 1. \textit{J. Almeida} and \textit{P. Weil} [Free profinite monoids: an introduction and examples, in J. Fountain (ed.), Semigroups, Formal languages and Groups, Kluwer Academic Publishers, 73-117 (1995)] have identified the topological semigroup \(\overline{F}_n{\mathcal V}\) with the projective limit of the projective system of all \(n\)-generated semigroups of the pseudovariety \(\mathcal V\). Since the ideal \(I_1{\mathcal S}\) is known to be nothing but the group \(\overline{F}_1{\mathcal G}\) of unary implicit operations in the pseudovariety \(\mathcal G\) of all finite groups, and the topological group \(\widehat{Z}\) is clearly isomorphic to the projective system of all cyclic groups, Theorem 1 of the paper under review is an obvious consequence of that general result. 2. The paper is badly translated. Among many mistakes, I mention the two most confusing only. The correct translation of the title of the paper should be ``Unary implicit operations on pseudo-varieties of semigroups''. The last sentence of the introduction translated as ``In the present article we essentially correct the theorem by J. Almeida and A. Azevedo \dots'' in fact should have been translated as ``In the present article we essentially refine a theorem by J. Almeida and A. Azevedo \dots''.