Relational Algebraic Approach to the Real Numbers: The Least-Upper-Bound Property

TL;DR

This paper explores the axiomatization of real numbers within relation algebra categories, focusing on the least-upper-bound property and extending previous work on their algebraic structure.

Contribution

It introduces a relation algebraic framework for the real numbers that incorporates the least-upper-bound property, building on prior axiomatizations.

Findings

Real number objects form a dense, linear ordered abelian group

The least-upper-bound property is established within the relation algebraic framework

Extension of axiomatization to include completeness property

Abstract

In this paper we continue the investigation of a real number object, i.e., an object representing the real numbers, in categories of relations. Our axiomatization is based on a relation algebraic version of Tarski's axioms of the real numbers. It was already shown that the addition of such an object forms a dense, linear ordered abelian group. In the current paper we will focus on the least-upper-bound property of such an object.