On profinite completions and canonical extensions
Let \(A\) be a (universal) algebra. Let \(\Phi\) be the set of all congruences \(\theta\) on \(A\) with \(A/\theta\) finite and let \(\bar A\) denote the subalgebra of elements \(\alpha\in \prod_{\Phi} A/\theta\) such that \(\alpha(\theta)=a/\theta\) and \(\theta\subseteq \theta_1\) implies \(\alpha/\theta_1=a/\theta_1.\) Then \(\bar A\) is called a profinite completion of \(A.\) In addition, a canonical extension of a bounded lattice \(L\) is a pair \((e, C)\), such that (i) \(C\) is a complete lattice, (ii) \(e: L\rightarrow C\) is a bounded lattice embedding, (iii) each element of \(C\) is a join of meets and a meet of joins of elements of the image \(e[L]\) of \(L\), and (iv) if \(F\), \(I\) are a filter and an ideal of \(L\), then \(\bigwedge e(F)\leq \bigvee e[I]\) implies \(F\cap I\neq \emptyset.\) Eventually, a bounded lattice \(L\) with a family of monotone fundamental operations (= \(n\)-ary operations that in each coordinate either preserve or reverse order) is called a monotone lattice expansion. The author proves: Let \(V\) be a variety of monotone lattice operations which is finitely generated. Then, profinite completions coincide with canonical extensions on \(V.\) Examples: Boolean algebras, bounded distributive lattices, or finitely generated varieties of Heyting algebras.
scientific article; zbMATH DE number 5119638