Disjoint $n$-amalgamation and pseudofinite countably categorical theories

DOI10.1215/00294527-2018-0025arXiv1510.03539MaRDI QIDQ6266367

Alex Kruckman

Publication date: 13 October 2015

Abstract: Disjoint $n$-amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this paper, we show that if a countably categorical theory $T$ admits an expansion with disjoint $n$-amalgamation for all $n$, then $T$ is pseudofinite. All theories which admit an expansion with disjoint $n$-amalgamation for all $n$ are simple, but the method can be extended, using filtrations of Fra"iss'e classes, to show that certain non-simple theories are pseudofinite. As case studies, we examine two generic theories of equivalence relations, $T_{extfeq}^{*}$ and $T_{extCPZ}$, and show that both are pseudofinite. The theories $T_{extfeq}^{*}$ and $T_{extCPZ}$ are not simple, but they are NSOP${}_1$. This is established here for $T_{extCPZ}$ for the first time.

Mathematics Subject Classification ID

Model theory of finite structures (03C13) Classification theory, stability, and related concepts in model theory (03C45) Categoricity and completeness of theories (03C35)

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