Relational Algebraic Approach to the Real Numbers the Additive Group

In this paper we start the investigation of an object representing the real numbers in categories of relations. Our axiomatization uses the construction of a relation power, i.e., an abstract version of power sets within the category. This allows us to utilize a relation algebraic version of Tarski's axioms of the real numbers as a first-order definition of a real number object. The current paper focuses on the addition operation of the real number object. It is shown that addition forms a densely and linearly ordered abelian group.