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A note on stable sets, groups, and theories with NIP
Author(s) -
Onshuus Alf,
Peterzil Ya'acov
Publication year - 2007
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200610046
Subject(s) - nip , mathematics , combinatorics , rank (graph theory) , set (abstract data type) , group (periodic table) , stability (learning theory) , discrete mathematics , physics , computer science , quantum mechanics , machine learning , programming language
Let M be an arbitrary structure. Then we say that an M ‐formula φ ( x ) defines a stable set in M if every formula φ ( x ) ∧ α ( x , y ) is stable. We prove: If G is an M ‐definable group and every definable stable subset of G has U ‐rank at most n (the same n for all sets), then G has a maximal connected stable normal subgroup H such that G / H is purely unstable. The assumptions hold for example if M is interpretable in an o‐minimal structure. More generally, an M ‐definable set X is weakly stable if the M ‐induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)