4 research outputs found
Definable group extensions in semi-bounded o-minimal structures
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
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
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 ) if every type over a sufficiently
rich base structure can be viewed as part of a -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 , 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
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