On the universal completions of pointfree function spaces
Imanol Mozo Carollo
TL;DR
This work develops a coherent pointfree framework for the universal completion of the Riesz space $\mathrm{C}(L)$ of continuous real functions on a completely regular frame $L$. It shows that the universal completion is realized naturally as $\mathrm{C}(\mathfrak{B}(L))$, via the embedding $\Upsilon(f)=\beta\cdot f$, and provides an equivalent, independent description through nearly finite Hausdorff continuous functions $\mathrm{H}_{nf}(L)$, which also yields a universal completion. The authors characterize when $\mathrm{C}(L)$ is universally complete (precisely for almost Boolean frames) and establish that two frames have isomorphic universal completions exactly when their Booleanizations are isomorphic. In the spatial case, they connect the pointfree constructions to classical representations, offering a new lattice-theoretic proof of van der Walt’s results and a pointfree Maeda-Ogasawara-Vulikh representation theorem that avoids Stone spaces and extended valued functions. Collectively, the paper provides new, constructive representations of universal completions and clarifies the role of Booleanization and nearly finite interval-valued functions in the theory of Archimedean Riesz spaces.
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.