Cuts of an ordered abelian group

Abstract

In this article, we study the cuts of a totally ordered abelian group $\Gamma$. We begin by recalling some results on ordered sets I and on the associated sets IS(I) and FS(I) of initial and final segments of I. For a totally ordered set I we review the notion of an I-structure defined on a module over a ring R, and the definition of the Hahn product of a family of R-modules indexed by I. The set Cv($\Gamma$)of convex subgroups of a totally ordered group $\Gamma$ is also a totally ordered set, canonically isomorphic to the set of cuts of the subset Pr($\Gamma$)of principal convex subgroups. One of the first results is then to equip the group $\Gamma$ with an I-structure where I is the set Pr($\Gamma$) endowed with the opposite order. We associate a convex subgroup with every cut of the group $\Gamma$, and conversely, we can associate a family of cuts with every convex subgroup of $\Gamma$. It is by looking at these subgroups, and the I-structure of $\Gamma$ that we can obtain a classification of the different types of cuts.