Simple-like independence relations in abstract elementary classes

DOI10.1016/J.APAL.2021.102971arXiv2003.02705MaRDI QIDQ6336191

Rami Grossberg, Marcos Mazari-Armida

Publication date: 5 March 2020

Abstract: We introduce and study simple and supersimple independence relations in the context of AECs with a monster model. $Theorem$: Let $K$ be an AEC with a monster model. - If $K$ has a simple independence relation, then $K$ does not have the 2-tree property. - If $K$ has a simple independence relation with $(<alep{h}_0)$-witness property, then $K$ does not have the tree property. The proof of both facts is done by finding cardinal bounds to classes of small Galois-types over a fixed model that are inconsistent for large subsets. We think this finer way of counting types is an interesting notion in itself. We characterize supersimple independence relations by finiteness of the Lascar rank under locality assumptions on the independence relation.

Mathematics Subject Classification ID

Classification theory, stability, and related concepts in model theory (03C45) Other infinitary logic (03C75) Set-theoretic model theory (03C55) Abstract elementary classes and related topics (03C48)

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