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# T-Convexity and Tame Extensions

Journal of Symbolic Logic 60 (1):74 - 102 (1995)

 Definable Choice for a Class of Weakly o-Minimal Theories.Michael C. Laskowski & Christopher S. Shaw - 2016 - Archive for Mathematical Logic 55 (5-6):735-748.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} (...)No categories Direct download (4 more)     Export citation Bookmark 1 citation Transseries and Todorov–Vernaeve’s Asymptotic Fields.Matthias Aschenbrenner & Isaac Goldbring - 2014 - Archive for Mathematical Logic 53 (1-2):65-87.We study the relationship between fields of transseries and residue fields of convex subrings of non-standard extensions of the real numbers. This was motivated by a question of Todorov and Vernaeve, answered in this paper. Direct download (5 more)     Export citation Bookmark Positive Primitive Formulae of Modules Over Rings of Semi-Algebraic Functions on a Curve.Laura R. Phillips - 2015 - Archive for Mathematical Logic 54 (5-6):587-614.Let R be a real closed field, and X⊆Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\subseteq R^m}$$\end{document} semi-algebraic and 1-dimensional. We consider complete first-order theories of modules over the ring of continuous semi-algebraic functions X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\to R}$$\end{document} definable with parameters in R. As a tool we introduce -piecewise vector bundles on X and show that the category of piecewise vector bundles on X is equivalent to the category of syzygies of (...) Direct download (2 more)     Export citation Bookmark The Elementary Theory of Dedekind Cuts in Polynomially Bounded Structures.Marcus Tressl - 2005 - Annals of Pure and Applied Logic 135 (1-3):113-134.Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of parameters from M, which yields a description of all subsets of Mn, definable in the expanded structure. Direct download (8 more)     Export citation Bookmark 1 citation Pseudo Completions and Completions in Stages of o-Minimal Structures.Marcus Tressl - 2006 - Archive for Mathematical Logic 45 (8):983-1009.For an o-minimal expansion R of a real closed field and a set $\fancyscript{V}$ of Th(R)-convex valuation rings, we construct a “pseudo completion” with respect to $\fancyscript{V}$ . This is an elementary extension S of R generated by all completions of all the residue fields of the $V \in \fancyscript{V}$ , when these completions are embedded into a big elementary extension of R. It is shown that S does not depend on the various embeddings up to an R-isomorphism. For polynomially (...) Direct download (5 more)     Export citation Bookmark 1 citation O-Minimal Residue Fields of o-Minimal Fields.Jana Maříková - 2011 - Annals of Pure and Applied Logic 162 (6):457-464.Let R be an o-minimal field with a proper convex subring V. We axiomatize the class of all structures such that , the corresponding residue field with structure induced from R via the residue map, is o-minimal. More precisely, in Maříková  it was shown that certain first-order conditions on are sufficient for the o-minimality of . Here we prove that these conditions are also necessary. Direct download (7 more)     Export citation Bookmark 1 citation Definable Functions Continuous on Curves in o-Minimal Structures.Janak Ramakrishnan - 2014 - Annals of Pure and Applied Logic 165 (7-8):1339-1351.We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve has a definable extension to a closed set. This situation is translated into a question about types: What are the conditions on an n-type such that, for any bounded definable function, the germ of the function on the type has a definable continuous global extension? Certain categories of (...) Direct download (3 more)     Export citation Bookmark Non-Archimedean Stratifications of Tangent Cones.Erick García Ramírez - 2017 - Mathematical Logic Quarterly 63 (3-4):299-312.We study the impact of a kind of non-archimedean stratifications on tangent cones of definable sets in real closed fields. We prove that such stratifications induce stratifications of the same nature on the tangent cone of a definable set at a fixed point. As a consequence, the archimedean counterpart of a t-stratification is shown to induce Whitney stratifications on the tangent cones of a semi-algebraic set. Extensions of these results are proposed for real closed fields with further structure. No categories Direct download (3 more)     Export citation Bookmark 