A lattice-theoretic approach to arbitrary real functions on frames
Imanol Mozo Carollo
TL;DR
This work develops a lattice-theoretic, pointfree framework to model arbitrary real-valued functions on frames by embedding extended semicontinuous functions into a Dedekind-MacNeille completion. It introduces extended lower/upper semicontinuous notions, characterizes the Dedekind completion of the lattice of continuous extended real functions, and defines the lattice $\mathrm{F}(L)$ of arbitrary extended real functions, along with the real-valued sublattice $\mathrm{F}(L) \cong \mathrm{C}(\reflectbox{$S$}(L))$ and the Booleanization connection $\mathrm{F}(L) \cong \mathrm{C}(\mathfrak{B}(\reflectbox{$S$}(L)))$. The paper proves that $\mathrm{F}(L)$ provides a conservative, order-theoretic extension of classical semicontinuous theories, with $F(L)$ behaving well under subfit and extremal-disconnectedness conditions, and it establishes a lattice-ordered-ring structure that supports a ring-of-real-functions viewpoint. It also clarifies the relationship to prior pointfree approaches (GKP2009, PP16), showing how the new notion specializes to classical cases for $T_1$ and $T_D$ spaces and when $S(L)$ is extremally disconnected. Overall, the results yield a robust, frame-theoretic model for discontinuous real functions with clear connections to classical topology and potential for further algebraic development of rings of real-valued functions on frames.
Abstract
In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if $L$ is a subfit frame, arbitrary extended real functions on $L$ are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuous functions on $L$. This approach mimicks the situation one has with a $T_1$-space $X$, where the lattice $\overline{\mathrm{F}}(X)$ of arbitrary extended real functions on $X$ is the smallest complete lattice containing both extended upper and lower semicontinuous functions on $X$. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for $T_1$-spaces. We also analyze semicontinuity and introduce definitions which are conservative for $T_D$-spaces.