2,227 research outputs found
Natural models of theories of green points
We explicitly present expansions of the complex field which are models of the
theories of green points in the multiplicative group case and in the case of an
elliptic curve without complex multiplication defined over . In
fact, in both cases we give families of structures depending on parameters and
prove that they are all models of the theories, provided certain instances of
Schanuel's conjecture or an analogous conjecture for the exponential map of the
elliptic curve hold. In the multiplicative group case, however, the results are
unconditional for generic choices of the parameters
Discrete subgroups of locally definable groups
We work in the category of locally definable groups in an o-minimal expansion
of a field. Eleftheriou and Peterzil conjectured that every definably generated
abelian connected group G in this category is a cover of a definable group. We
prove that this is the case under a natural convexity assumption inspired by
the same authors, which in fact gives a necessary and sufficient condition. The
proof is based on the study of the zero-dimensional compatible subgroups of G.
Given a locally definable connected group G (not necessarily definably
generated), we prove that the n-torsion subgroup of G is finite and that every
zero-dimensional compatible subgroup of G has finite rank. Under a convexity
hypothesis we show that every zero-dimensional compatible subgroup of G is
finitely generated.Comment: Final version. 17 pages. To appear in Selecta Mathematic
DEFINABLE SETS IN DP-MINIMAL ORDERED ABELIAN GROUPS (Model theoretic aspects of the notion of independence and dimension)
This article surveys some recent results on ordered abelian groups (possibly with additional definable structure) from the subclass of NIP theories which are dp-minimal. To put these results in context, the first part of the article reviews and compares various other generalizations of a-minimality (such as local a-minimality and a-stability) and their consequences. It is useful to make the further assumption that there is a cardinal bound on the number of convex subgroups definable in elementary extensions of the structure. Under this hypothesis, some classic theorems on o-minimal structures, such as the monotonicity theorem for unary definable functions, can be suitably generalized
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