A Point-Free Approach to Canonical Extensions of Boolean Algebras and Bounded Archimedean $$\ell$$-Algebras

Abstract

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 $$\mathcal {L}_B$$ generated by the bounded meet-semilattice of all filters of B (ordered by reverse inclusion) and prove that the booleanization of $$\mathcal {L}_B$$ is a canonical extension of B. Our main result generalizes this approach to the category $$\varvec{ ba \ell }$$ of bounded archimedean $$\ell$$ -algebras, thus yielding a point-free construction of canonical extensions in $$\varvec{ ba \ell }$$ . We conclude by showing that the algebra of normal functions on the Alexandroff space of proper archimedean $$\ell$$ -ideals of A is a canonical extension of $$A\in \varvec{ ba \ell }$$ , thus providing a generalization of the result of Holliday to $$\varvec{ ba \ell }$$ .