Arithmetic of Dedekind cuts of ordered Abelian groups

Abstract

We study the set of Dedekind cuts over a linearly ordered Abelian group as a structure over the language (0,<,+,-). Moreover, we obtain a simple set of axioms for the universal part of the theory of such structures. Finally, we prove that every structure satisfying the given axioms is a sub-structure of the set of cuts over a suitable group.