Natural extensions and profinite completions of algebras

Algebra Universalis, 66(3) (2011), 205-241
http://dx.doi.org/10.1007/s00012-011-0155-y

This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class TeX , where TeX is a set, not necessarily finite, of finite algebras, it is shown that each TeX embeds as a topologically dense subalgebra of a topological algebra TeX (its natural extension), and that TeX 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 TeX is finite and TeX 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.