A Point-Free Approach to Canonical Extensions of Boolean Algebras and Bounded Archimedean ℓ -Algebras

Recently W. Holliday gave a choice-free construction of a canonical extension of a boolean algebra B as the boolean algebra of regular open subsets of the Alexandroff topology on the poset of proper filters of B. We make this construction point-free by replacing the Alexandroff space of proper filters of B with the free frame LB generated by the bounded meet-semilattice of all filters of B (ordered by reverse inclusion) and prove that the booleanization of LB is a canonical extension of B. Our main result generalizes this approach to the category baℓ of bounded archimedean ℓ-algebras, thus yielding a point-free construction of canonical extensions in baℓ. We conclude by showing that the algebra of normal functions on the Alexandroff space of proper archimedean ℓ-ideals of A is a canonical extension of A∈ baℓ, thus providing a generalization of the result of Holliday to baℓ.