Cuts of an ordered abelian group
Michel Vaquié
TL;DR
The paper surveys the structure of cuts and initial/final segments in totally ordered abelian groups, emphasizing their central role in valuation theory and the behavior of value groups.It develops a unified framework using I-structures and Hahn-products to relate filtrations on groups and modules to associated graded data, and then applies this to convex-subgroup lattices and principal subgroups.A detailed classification of cuts is provided via invariance subgroups and symmetric-interval analysis, along with a careful study of how cuts behave under morphisms and quotients, including notions of well-ranked, discrete, and rank-one behavior.Overall, the work connects order-theoretic constructions (cuts and segments) with algebraic structures (convex subgroups, Hahn-products, and divisibility) to enable a systematic understanding of valuations and their extensions.
Abstract
In this article, we study the cuts of a totally ordered abelian group $Γ$. 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($Γ$)of convex subgroups of a totally ordered group $Γ$ is also a totally ordered set, canonically isomorphic to the set of cuts of the subset Pr($Γ$)of principal convex subgroups. One of the first results is then to equip the group $Γ$ with an I-structure where I is the set Pr($Γ$) endowed with the opposite order. We associate a convex subgroup with every cut of the group $Γ$, and conversely, we can associate a family of cuts with every convex subgroup of $Γ$. It is by looking at these subgroups, and the I-structure of $Γ$ that we can obtain a classification of the different types of cuts.