Defines k-trace definability and proves that theories k-trace definable in NIP theories are k-NIP, with universality constructions for main examples like Hilbert space and generic hypergraphs.
Hodges.Model theory, volume 42 ofEncyclopedia of mathematics and its applications
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A weakly o-minimal structure without infinite groups has a Shelah completion that interprets an infinite field, with a new local trace definability notion and a linearity-field dichotomy for certain ordered groups.
For a number of theories T* of model-theoretic interest there exists a simpler theory T and infinite cardinal κ such that T* is trace equivalent to the theory of κ-dimensional space over a model of T.
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Trace definability IV: higher arity notions
Defines k-trace definability and proves that theories k-trace definable in NIP theories are k-NIP, with universality constructions for main examples like Hilbert space and generic hypergraphs.
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Trace definability II: model-theoretic linearity
A weakly o-minimal structure without infinite groups has a Shelah completion that interprets an infinite field, with a new local trace definability notion and a linearity-field dichotomy for certain ordered groups.
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Trace definability III: Infinite dimensional space over a model of $T$
For a number of theories T* of model-theoretic interest there exists a simpler theory T and infinite cardinal κ such that T* is trace equivalent to the theory of κ-dimensional space over a model of T.