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August 2022 On Stable Quotients
Krzysztof Krupiński, Adrián Portillo
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Notre Dame J. Formal Logic 63(3): 373-394 (August 2022). DOI: 10.1215/00294527-2022-0023

Abstract

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then Gst=G00 (where Gst is the smallest ∧-definable subgroup with GGst stable, and G00 is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion Theq. The second result is an example of a group G definable in a nondistal NIP theory for which G=G00, but Gst is not an intersection of definable groups. Our example is a saturated extension of (R,+,[0,1]). Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic.

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Krzysztof Krupiński. Adrián Portillo. "On Stable Quotients." Notre Dame J. Formal Logic 63 (3) 373 - 394, August 2022. https://doi.org/10.1215/00294527-2022-0023

Information

Received: 6 October 2021; Accepted: 26 April 2022; Published: August 2022
First available in Project Euclid: 25 September 2022

Digital Object Identifier: 10.1215/00294527-2022-0023

Subjects:
Primary: 03C45

Keywords: distal theory , hyperimaginary , model-theoretic components , stable quotient

Rights: Copyright © 2022 University of Notre Dame

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Vol.63 • No. 3 • August 2022
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