1,395 research outputs found
Beautiful pairs
We introduce an abstract framework to study certain classes of stably
embedded pairs of models of a complete -theory , called
beautiful pairs, which comprises Poizat's belles paires of stable structures
and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an
amalgamation construction, we relate several properties of beautiful pairs with
classical Fra\"{i}ss\'{e} properties.
After characterizing beautiful pairs of various theories of ordered abelian
groups and valued fields, including the theories of algebraically, -adically
and real closed valued fields, we show an Ax-Kochen-Ershov type result for
beautiful pairs of henselian valued fields. As an application, we derive strict
pro-definability of particular classes of definable types. When is one of
the theories of valued fields mentioned above, the corresponding classes of
types are related to classical geometric spaces such as Berkovich and Huber's
analytifications. In particular, we recover a result of Hrushovski-Loeser on
the strict pro-definability of stably dominated types in algebraically closed
valued fields.Comment: 40 page
External definability and groups in NIP theories
We prove that many properties and invariants of definable groups in NIP
theories, such as definable amenability, G/G^{00}, etc., are preserved when
passing to the theory of the Shelah expansion by externally definable sets,
M^{ext}, of a model M. In the light of these results we continue the study of
the "definable topological dynamics" of groups in NIP theories. In particular
we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in
some new cases, including definably amenable groups in o-minimal structures.Comment: 28 pages. Introduction was expanded and some minor mistakes were
corrected. Journal of the London Mathematical Society, accepte
Generic Automorphisms and Green Fields
We show that the generic automorphism is axiomatisable in the green field of
Poizat (once Morleyised) as well as in the bad fields which are obtained by
collapsing this green field to finite Morley rank. As a corollary, we obtain
"bad pseudofinite fields" in characteristic 0. In both cases, we give geometric
axioms. In fact, a general framework is presented allowing this kind of
axiomatisation. We deduce from various constructibility results for algebraic
varieties in characteristic 0 that the green and bad fields fall into this
framework. Finally, we give similar results for other theories obtained by
Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories
having the definable multiplicity property. We also close a gap in the
construction of the bad field, showing that the codes may be chosen to be
families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap
in the construction of the bad fiel
Definability in the embeddability ordering of finite directed graphs, II
We deal with first-order definability in the embeddability ordering of finite directed graphs. A directed graph is said to be embeddable into if there exists
an injective graph homomorphism . We describe the
first-order definable relations of using the first-order
language of an enriched small category of digraphs. The description yields the
main result of one of the author's papers as a corollary and a lot more. For
example, the set of weakly connected digraphs turns out to be first-order
definable in . Moreover, if we allow the usage of a
constant, a particular digraph , in our first-order formulas, then the full
second-order language of digraphs becomes available
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