A point-free approach to canonical extensions of boolean algebras and bounded archimedean $\ell$-algebras

TL;DR

This paper develops a point-free, algebraic construction of canonical extensions for boolean algebras and bounded archimedean ll-algebras, generalizing previous topological approaches.

Contribution

It introduces a new point-free method for constructing canonical extensions, extending the approach from boolean algebras to bounded archimedean ll-algebras.

Findings

The construction of canonical extensions for boolean algebras is generalized to ll-algebras.

The algebra of normal functions on the Alexandroff space forms a canonical extension.

The approach simplifies and unifies the understanding of canonical extensions in these algebraic structures.

Abstract

In \cite{BH20} an elegant choice-free construction of a canonical extension of a boolean algebra $B$ was given 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 $\mathcal{L}$ generated by the bounded meet-semilattice of all filters of $B$ (ordered by reverse inclusion) and prove that the booleanization of $\mathcal{L}$ is a canonical extension of $B$. Our main result generalizes this approach to the category $\boldsymbol{\mathit{ba}\ell}$ of bounded archimedean $\ell$-algebras, thus yielding a point-free construction of canonical extensions in $\boldsymbol{\mathit{ba}\ell}$. 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 $A\in\boldsymbol{\mathit{ba}\ell}$, thus providing a generalization of the result of \cite{BH20} to $\boldsymbol{\mathit{ba}\ell}$.