Extending maps to profinite completions in finitely generated quasivarieties
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Extending maps to profinite completions in finitely generated quasivarieties Georges Hansoul1 · Bruno Teheux2 Received: 5 June 2019 / Accepted: 19 February 2020 © The Managing Editors 2020
Abstract We consider the problem of extending maps from algebras to their profinite completions in finitely generated quasivarieties. Our developments are based on the construction of the profinite completion of an algebra as its natural extension. We provide an extension which is a multi-map and we study its continuity properties, and the conditions under which it is a map. Keywords Profinite completions · Natural dualities · Natural extensions · Canonical extensions Mathematics Subject Classification 03C05 · 08C20
1 Introduction This paper is a contribution to the study of profinite completions in internally residually finite prevarieties. A class A of algebras is called (Davey et al. 2011) an internally residually finite prevariety (IRF-prevariety for short) if there is a set M of finite algebras such that A = ISP(M). Every algebra A of an IRF-prevariety A embeds in its A-profinite completion proA (A), which is defined as the inverse limit of the inverse system of the finite quotients of A that belongs to A, with natural homomorphisms as bonding maps (see Sect. 2 for details). In what follows, we limit ourself to those IRF-prevareties A that are finitely generated quasivarieties, i.e., for which there is a finite set M of finite algebras such that A = ISP(M). Moreover, we assume
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Bruno Teheux [email protected] Georges Hansoul [email protected]
1
Département de Mathématiques, Université de Liège, Grande Traverse, 12, 4000 Liège, Belgium
2
Department of Mathematics, FSTM, University of Luxembourg, 6, Avenue de la Fonte, 4364 Esch-sur-Alzette, Luxembourg
123
Beitr Algebra Geom
that A = ISP({M}), but this restriction is a matter of convenience: we claim that our developments admit the obvious generalization to the multi-sorted case where M = {M1 , . . . , Mn }. It is proved in Davey et al. (2011) that proA (A) is isomorphic to the natural ∗ extension Aδ of A, that is, the topological closure of eA (A) in MιA , where A∗ = ∗ A(A, M), where eA : A → MA is the evaluation map defined as eA (a)(φ) = φ(a), and where ι is the discrete topology on M (this representation result actually holds in any IRF-prevariety). Moreover, if M is a discrete structure that yields a natural as a (closed) substructure of M A , then Aδ can duality for A, and if A∗ is considered from A∗ to M be concretely computed as the algebra of structure preserving maps (Davey et al. 2011, Theorem 4.3). With these results in mind, we adopt the notation δ A to denote proA (A) for the remaining of the paper. We consider the following problem: given A, B ∈ A and a map u : A → B, how to define a ‘reasonable’ extension u δ : Aδ → Bδ of u? Such an extension would allow to study profinite completions of expansions of A-algebras, and preservation of equations through profinite completions. This problem has a well known solution (Gehrke and Jónsson 2004) in
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