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MAXIMAL STABLE QUOTIENTS OF INVARIANT TYPES IN NIP THEORIES

Part of: Model theory

Published online by Cambridge University Press:  25 October 2023

KRZYSZTOF KRUPIŃSKI Open the ORCID record for KRZYSZTOF KRUPIŃSKI [Opens in a new window]  and
ADRIÁN PORTILLO Open the ORCID record for ADRIÁN PORTILLO [Opens in a new window]
KRZYSZTOF KRUPIŃSKI*
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTETU WROCŁAWSKIEGO PL. GRUNWALDZKI 2, 50-384 WROCŁAW, POLAND E-mail: Adrian.Portillo-Fernandez@math.uni.wroc.pl
ADRIÁN PORTILLO
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTETU WROCŁAWSKIEGO PL. GRUNWALDZKI 2, 50-384 WROCŁAW, POLAND E-mail: Adrian.Portillo-Fernandez@math.uni.wroc.pl
*
E-mail: Krzysztof.Krupinski@math.uni.wroc.pl
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Abstract

For a NIP theory T, a sufficiently saturated model ${\mathfrak C}$ of T, and an invariant (over some small subset of ${\mathfrak C}$) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from ${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main result of [2] on the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in [3].

Keywords

stable quotienthyperimaginaryinvariant type

MSC classification

Primary: 03C45: Classification theory, stability and related concepts
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 25
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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