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December 2010 Grouplike minimal sets in ACFA and in TA
Alice Medvedev
J. Symbolic Logic 75(4): 1462-1488 (December 2010). DOI: 10.2178/jsl/1286198157

Abstract

This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in TA. The thesis concerns minimal formulae of the form x ∈ A ∧ σ(x) = f(x) for an algebraic curve A and a dominant rational function f: A → σ(A). These are shown to be uniform in the Zilber trichotomy, and the pairs (A,f) that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to TA, the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A TA analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry.

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Alice Medvedev. "Grouplike minimal sets in ACFA and in TA." J. Symbolic Logic 75 (4) 1462 - 1488, December 2010. https://doi.org/10.2178/jsl/1286198157

Information

Published: December 2010
First available in Project Euclid: 4 October 2010

zbMATH: 1235.03069
MathSciNet: MR2767979
Digital Object Identifier: 10.2178/jsl/1286198157

Rights: Copyright © 2010 Association for Symbolic Logic

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Vol.75 • No. 4 • December 2010
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