4 research outputs found

    Definable group extensions in semi-bounded o-minimal structures

    Get PDF
    In this note we show: Let ℛ = 〈 R, <, +, 0,...〉 be a semi-bounded (respectively, linear) o-minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈 Rm, +〉.FCT Financiamento Base 2008 - USFL/1/209; FCT grant SFRH/BPD/35000/200

    Coverings by open cells

    Full text link
    We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells.Comment: 17 pages, revised versio

    Valued fields, Metastable groups

    Full text link
    We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (over a sort Γ\Gamma) if every type over a sufficiently rich base structure can be viewed as part of a Γ\Gamma-parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from Γ\Gamma, and a definable direct limit system of groups with stably dominated generic. In the case of ACVF, among definable subgroups of affine algebraic groups, we characterize the groups with stably dominated generics in terms of group schemes over the valuation ring. Finally, we classify all fields definable in ACVF.Comment: 48 pages. Minor corrections and improvements following a referee repor
    corecore