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A. Pillay (2003)
On countable simple unidimensional theoriesJournal of Symbolic Logic, 68
Byunghan Kim, A. Pillay (1997)
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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
Ziv Shami (2013)
On uncountable hypersimple unidimensional theoriesArchive for Mathematical Logic, 53
Ziv Shami (2011)
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Let T be a simple L-theory and let T-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T^-$$\end{document} be a reduct of T to a sublanguage L-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^-$$\end{document} of L. For variables x, we call an ∅\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\emptyset $$\end{document}-invariant set Γ(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma (x)$$\end{document} in C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}$$\end{document} a universal transducer if for every formula ϕ-(x,y)∈L-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi ^-(x,y)\in L^-$$\end{document} and every a, ϕ-(x,a)L--forksover∅iffΓ(x)∧ϕ-(x,a)L-forksover∅.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \phi ^-(x,a)\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma (x)\wedge \phi ^-(x,a)\ L\text{-forks } \text{ over }\ \emptyset . \end{aligned}$$\end{document}We show that there is a greatest universal transducer Γ~x\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{\Gamma }_x$$\end{document} (for any x) and it is type-definable. In particular, the forking topology on Sy(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_y(T)$$\end{document} refines the forking topology on Sy(T-)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_y(T^-)$$\end{document} for all y. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that Γ~x\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{\Gamma }_x$$\end{document} is the unique universal transducer that is L-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^-$$\end{document}-type-definable with parameters. If T-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T^-$$\end{document} is a theory with the wnfcp (the weak nfcp) and T is the theory of its lovely pairs of models we show that Γ~x=(x=x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{\Gamma }_x=(x=x)$$\end{document} and give a more precise description of the set of universal transducers for the special case where T-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T^-$$\end{document} has the nfcp.
Archive for Mathematical Logic – Springer Journals
Published: May 29, 2020
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