Archive for Mathematical Logic 55 (5-6):735-748 (2016)

Abstract
Given an o-minimal structure M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}$$\end{document} with a group operation, we show that for a properly convex subset U, the theory of the expanded structure M′=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}'=$$\end{document} has definable Skolem functions precisely when M′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}'$$\end{document} is valuational. As a corollary, we get an elementary proof that the theory of any such M′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}'$$\end{document} does not satisfy definable choice.
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DOI 10.1007/s00153-016-0490-y
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References found in this work BETA

Weakly o-Minimal Nonvaluational Structures.Roman Wencel - 2008 - Annals of Pure and Applied Logic 154 (3):139-162.
T-Convexity and Tame Extensions II.Lou Van Den Dries - 1997 - Journal of Symbolic Logic 62 (1):14-34.
T-Convexity and Tame Extensions.Dries Lou Van Den & H. Lewenberg Adam - 1995 - Journal of Symbolic Logic 60 (1):74 - 102.
Paires de Structures o-Minimales.Yerzhan Baisalov & Bruno Poizat - 1998 - Journal of Symbolic Logic 63 (2):570-578.
Algebraic Theories with Definable Skolem Functions.Lou van den Dries - 1984 - Journal of Symbolic Logic 49 (2):625-629.

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