Decidability Results for Classes of Ordered Abelian Groups in Logics with Ramsey-quantifiers

This paper is to contributive to the model theory of ordered abelian groups (o.a.g. for short). The basic elements to build up the algebraic structure of the o.a.g-s are the archimedean groups: By Hahn's embedding theorem every o.a.g. can be represented as a subgroup of the Hahn-product of archimedean o.a.g.s. Archimedean is not a first-order concept but there exists a first-order model theory of o.a.g.-s has to be developed inside the framework of regularly o.a.g-s. Weispfenning [2] showed that these are essentially the ones that admit elimination of quantifiers in the language {+, −, 0, <, ≡ n |n < ω} of o.a.g-s. (possibly extended by a set of constant symbols). Moreover, this quantifier elimination procedure is a basic tool for the model-theoretic investigations in this field. We start to develop the model theory of o.a.g-s inside the framework of extended logics: archimedean is a L(Q n 0)-phenomenon Q n 0 being the " Ramsey quantifier " (in the ℵ 0-interpretation) introduced by Magidor & Malitz [1]. We generalize the quantifier elimination results mentioned above to L(Q n 0) and L <ω 0. Especially, we prove quantifier elimination results for classes of non-archimedean o.a.g.s. Since all these quantifier-elimination procedures are effective, this yields the decidability of the respective theories.