On the forking topology of a reduct of a simple theory

Archive for Mathematical Logic 59 (3-4):313-324 (2020)

Abstract

Let T be a simple L-theory and let \ be a reduct of T to a sublanguage \ of L. For variables x, we call an \-invariant set \\) in \ a universal transducer if for every formula \\in L^-\) and every a, $$\begin{aligned} \phi ^-\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma \wedge \phi ^-\ L\text{-forks } \text{ over }\ \emptyset. \end{aligned}$$We show that there is a greatest universal transducer \ and it is type-definable. In particular, the forking topology on \\) refines the forking topology on \\) for all y. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that \ is the unique universal transducer that is \-type-definable with parameters. If \ is a theory with the wnfcp and T is the theory of its lovely pairs of models we show that \\) and give a more precise description of the set of universal transducers for the special case where \ has the nfcp.

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References found in this work

Simple Theories.Byunghan Kim & Anand Pillay - 1996 - Annals of Pure and Applied Logic 88 (2):149-164.
Lovely Pairs of Models.Itay Ben-Yaacov, Anand Pillay & Evgueni Vassiliev - 2003 - Annals of Pure and Applied Logic 122 (1-3):235-261.
On Analyzability in the Forking Topology for Simple Theories.Ziv Shami - 2006 - Annals of Pure and Applied Logic 142 (1):115-124.
On Uncountable Hypersimple Unidimensional Theories.Ziv Shami - 2014 - Archive for Mathematical Logic 53 (1-2):203-210.

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Citations of this work

A Note on the Non‐Forking‐Instances Topology.Ziv Shami - 2020 - Mathematical Logic Quarterly 66 (3):336-340.

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