Natural extensions and profinite completions of algebras

Abstract

This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class A = ISP(M), where M is a set, not necessarily finite, of finite algebras, it is shown that each A ∈ A embeds as a topologically dense subalgebra of a topological algebra n A (A) (its natural extension), and that n A (A) is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that M is finite and A possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply.

Bohr Compactifications of Algebras and Structures

“…We can provide a quite explicit, if unwieldy, description of the elements of the natural extension in the context of an IRF-prevariety of structures. This generalises the description given by [17,Theorem 4.1] and is proved in the same way. We present this in the single-sorted case (so that |M| = 1) since this covers our future needs in this paper and simplifies the statement; a multi-sorted version could be obtained, as in [17].…”

“…We shall need this result shortly in order to prove that the natural extension functor is a reflection. The lemma extends to the setting of CT-prevarieties of structures an analogous result for IRFprevarieties of algebras given in [17]. We adopt the same notation as above.…”

“…It is well known, and has been proved in various ways (see [5,17,25]), that for a bounded distributive lattice L we have n D (L) ∼ = pro D (L) ∼ = L δ , where pro D (L) and L δ denote respectively the profinite completion and the canonical extension of L. It follows that the Priestley dual space of L δ is β (D(L) ♭ ). This recaptures [7,Corollary 5.4] (see also [5,Proposition 3.4]).…”

“…The following theorem, as it applies to an IRF-prevariety of algebras, appears in [17,Theorem 4.3]. The proof given in [17] extends immediately to total structures but not to structures in general. When a duality for A is known, Theorem 5.1 describes the elements of n A (A), which is defined topologically, in a way which is not overtly topological.…”

“…The natural extension construction has an important virtue. The formalism of traditional natural duality theory, as presented in the text of Clark and Davey [8], enables an explicit description to be given in general of n A (A), for A in an IRF-prevariety A of algebras, and in a more refined and amenable form, for classes which admit a natural duality [17]. Drawing similarly on duality theory ideas that extend to IRF-prevarieties of structures [16] and to certain CT-prevarieties of structures [20], we are able explicitly to describe natural extensions in the wider setting, though there are impediments: structures must not contain partial operations and, for prevarieties with infinite generators, the results we obtain are less complete than those for IRF-prevarieties.…”