Non-forking and preservation of NIP and dp-rank

Pedro Andrés Estevan; Itay Kaplan

doi:10.1016/j.apal.2021.102946

Non-forking and preservation of NIP and dp-rank

Pedro Andrés Estevan, Itay Kaplan^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We investigate the question of whether the restriction of an NIP type p∈S(B) which does not fork over A⊆B to A is also NIP, and the analogous question for dp-rank. We show that if B contains a Morley sequence I generated by p over A, then p↾AI is NIP and similarly preserves the dp-rank. This yields positive answers for generically stable NIP types and the analogous case of stable types. With similar techniques we also provide a new more direct proof for the latter. Moreover, we introduce a general construction of “trees whose open cones are models of some theory” and in particular an inp-minimal theory DTR of dense trees with random graphs on open cones, which exemplifies a negative answer to the question.

Original language	English
Article number	102946
Pages (from-to)	1-30
Number of pages	30
Journal	Annals of Pure and Applied Logic
Volume	172
Issue number	6
DOIs	https://doi.org/10.1016/j.apal.2021.102946
State	Published - Jun 2021

Bibliographical note

Funding Information:
The collaboration started during a research stay of the first author in The Hebrew University of Jerusalem partially supported by Fundaci?n Montcelimar, MTM2017-86777-P and 2017SGR-270.The second author would like to thank The Israel Science Foundation for their support of this research (Grants no. 1533/14 and 1254/18).

Publisher Copyright:
© 2021 Elsevier B.V.

Keywords

Forking
NIP/stable types
Trees
dp-Rank

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10.1016/j.apal.2021.102946

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Non-forking and preservation of NIP and dp-rank

Abstract

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Keywords

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