Extended real number arithmetics via Dedekind cuts

TL;DR

This paper develops a framework for extended real numbers using Dedekind cuts, establishing sound arithmetic laws and structures that unify proper and improper functions, with applications in set-valued analysis.

Contribution

It introduces a novel approach to extended real numbers via Dedekind cuts, defining two types of addition and complete lattice structures, enabling unified treatment of extended real-valued functions.

Findings

Defines inf-addition and sup-addition for extended reals

Establishes conlinear spaces that are complete lattices

Provides formulas for pseudo-differences like (+∞) - (+∞)

Abstract

It is shown how Dedekind cuts can be used to introduce the extended real numbers along with sound arithmetic laws via one simple rule for the addition of sets. The crucial idea is that the use of the lower and the upper part of the cuts, respectively, leads to two different additions which are known in the literature as inf-addition and sup-addition. Moreover, the two resulting structures are conlinear spaces which at the same time are complete lattices with respect to the natural order. This admits the definition of pseudo-differences on the extended reals which also provide formulas for expressions like $(+\infty) - (+\infty)$, $(-\infty) - (-\infty)$. There are two major motivations: one is that proper and improper extended real-valued functions can be treated in a unified manner, the other that set-valued functions can often be represented by families of scalar functions which may include improper ones.