Bounded orbits and strongly generic sets | Zendy

Newelski Ludomir | Zendy

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Bounded orbits and strongly generic sets

Author(s) -

Newelski Ludomir

Publication year - 2012

Publication title -

journal of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.441

H-Index - 62

eISSN - 1469-7750

pISSN - 0024-6107

DOI - 10.1112/jlms/jdr075

Subject(s) - bounded function , isomorphism (crystallography) , mathematics , orbit (dynamics) , group (periodic table) , space (punctuation) , action (physics) , ideal (ethics) , combinatorics , type (biology) , discrete mathematics , pure mathematics , mathematical analysis , computer science , physics , quantum mechanics , ecology , philosophy , chemistry , epistemology , crystal structure , engineering , biology , crystallography , aerospace engineering , operating system

Let G be a group definable in a theory T . We study the action of G on the space of its external types. We introduce the notion of a strongly generic subset of G . We prove that there is a bounded orbit of external types if and only if there are boundedly many externally definable strongly generic subsets of G . In this situation, we prove the following. The group G 00 exists and equals G ∞ , also the size of every bounded orbit of an external almost periodic type is bounded by 2 ℵ0 . The ideal subgroups of the space of external types of G up to isomorphism do not depend on the choice of a model of T . If additionally T is weakly o‐minimal, then G is definably amenable.

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