Natural extensions and profinite completions of algebras. (English) Zbl 1232.08006
A profinite completion of a given algebra \(A\) is defined to be the inverse limit algebra formed from finite quotient algebras of \(A\) by congruences of finite index, where the indexing set is the set of congruences of finite index and ordering is reverse inclusion. The paper investigates profinite completions of residually finite algebras drawing on ideas of natural dualities. Given a class \(V=\mathrm{ISP}(M)\), where \(M\) is a set of finite algebras, it is shown that each \(A \in V\) embeds as a topologically dense subalgebra of a topological algebra which is a natural extension of \(A\) and this algebra is isomorphic, topologically and algebraically, to the profinite completion of \(A\). It is shown how the natural extensions can be described as families of relation-preserving maps. For an algebra of a finitely generated variety of lattice-based algebras, the natural extension provides a realization of the canonical extension. The paper concludes with an exhaustive survey of classes of algebras to which the main theorem do, or do not, apply.
Reviewer: Ivan Chajda (Přerov)
MSC:
| 08B25 | Products, amalgamated products, and other kinds of limits and colimits |
| 06D50 | Lattices and duality |
| 08C15 | Quasivarieties |
| 08C20 | Natural dualities for classes of algebras |
| 18A30 | Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) |
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