On the universal completions of pointfree function spaces

TL;DR

This paper explores the universal completion of Riesz spaces of continuous functions on frames, offering new representations via Booleanization and nearly finite Hausdorff functions, and characterizing when these spaces are universally complete.

Contribution

It introduces novel pointfree constructions of universal completions of Riesz spaces, including representations through Booleanization and nearly finite Hausdorff functions, and characterizes universal completeness in this context.

Findings

Universal completions can be represented as spaces of continuous functions on Booleanizations.

C(L) and C(M) have isomorphic universal completions iff their Booleanizations are isomorphic.

Characterization of frames L for which C(L) is universally complete as almost Boolean frames.

Abstract

This paper approaches the construction of the universal completion of the Riesz space $\mathrm{C}(L)$ of continuous real functions on a completely regular frame $L$ in two different ways. Firstly as the space of continuous real functions on the Booleanization of $L$. Secondly as the space of nearly finite Hausdorff continuous functions on $L$. The former has no counterpart in the classical theory, as the Booleanization of a spatial frame is not spatial in general, and it offers a lucid way of representing the universal completion as a space of continuous real functions. As a corollary we obtain that $\mathrm{C}(L)$ and $\mathrm{C}(M)$ have isomorphic universal completions if and only if the Booleanization of $L$ and $M$ are isomorphic and we characterize frames $L$ such that $\mathrm{C}(L)$ is universally complete as almost Boolean frames. The application of this last result to the classical case $\mathrm{C}(X)$ of the space of continuous real functions on a topological space $X$ characterizes those spaces $X$ for which $\mathrm{C}(X)$ is universally complete. Finally, we present a pointfree version of the Maeda-Ogasawara-Vulikh representation theorem and use it to represent the universal completion of an Archimedean Riesz space with weak unit as a space of continuous real functions on a Boolean frame.