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Definable choice for a class of weakly o-minimal theories

Michael C. Laskowski & Christopher S. Shaw

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

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Weakly o-minimal nonvaluational structures.Roman Wencel - 2008 - Annals of Pure and Applied Logic 154 (3):139-162.details A weakly o-minimal structure image expanding an ordered group is called nonvaluational iff for every cut left angle bracketC,Dright-pointing angle bracket of definable in image, we have that inf{y−x:xset membership, variantC,yset membership, variantD}=0. The study of nonvaluational weakly o-minimal expansions of real closed fields carried out in [D. Macpherson, D. Marker, C. Steinhorn,Weakly o-minimal structures and real closed fields, Trans. Amer. Math. Soc. 352 5435–5483. MR1781273 (2001i:03079] suggests that this class is very close to the class of o-minimal expansions of (...) Model Theory in Logic and Philosophy of Logic Science, Logic, and Mathematics Direct download (5 more) Export citation Bookmark 13 citations
Tame Topology and O-Minimal Structures.Lou van den Dries - 2000 - Bulletin of Symbolic Logic 6 (2):216-218.details Logic and Philosophy of Logic Model Theory in Logic and Philosophy of Logic Direct download Export citation Bookmark 41 citations
$t$-convexity And Tame Extensions.Lou van den Dries & Adam H. Lewenberg - 1995 - Journal of Symbolic Logic 60 (1):74-102.details Let $T$ be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of $T$ and show that the residue field of such a convex hull has a natural expansion to a model of $T$. We give a quantifier elimination relative to $T$ for the theory of pairs $$ where $\mathscr{R} \models T$ and $V \neq \mathscr{R}$ is the convex hull of an elementary substructure of $\mathscr{R}$. We deduce that (...) Logic and Philosophy of Logic Direct download (2 more) Export citation Bookmark 17 citations
T-convexity and tame extensions II.Lou van den Dries - 1997 - Journal of Symbolic Logic 62 (1):14-34.details I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on its (...) Logic and Philosophy of Logic Model Theory in Logic and Philosophy of Logic Direct download (6 more) Export citation Bookmark 11 citations
Algebraic theories with definable Skolem functions.Lou van den Dries - 1984 - Journal of Symbolic Logic 49 (2):625-629.details Logic and Philosophy of Logic Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic Model Theory in Logic and Philosophy of Logic Direct download (5 more) Export citation Bookmark 14 citations
Omitting types in o-minimal theories.David Marker - 1986 - Journal of Symbolic Logic 51 (1):63-74.details Logic and Philosophy of Logic Model Theory in Logic and Philosophy of Logic Direct download (6 more) Export citation Bookmark 8 citations
T-Convexity and Tame Extensions II.Lou Van Den Dries - 1997 - Journal of Symbolic Logic 62 (1):14 - 34.details I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on its (...) Logic and Philosophy of Logic Model Theory in Logic and Philosophy of Logic Direct download (7 more) Export citation Bookmark 12 citations
T-Convexity and Tame Extensions.Dries Lou Van Den & H. Lewenberg Adam - 1995 - Journal of Symbolic Logic 60 (1):74 - 102.details Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (R, V) where $\mathscr{R} \models T$ and V ≠ R is the convex hull of an elementary substructure of R. We deduce (...) Logic and Philosophy of Logic Model Theory in Logic and Philosophy of Logic Direct download (9 more) Export citation Bookmark 17 citations
Algebraic Theories with Definable Skolem Functions.Lou van Den Dries - 1984 - Journal of Symbolic Logic 49 (2):625 - 629.details Logic and Philosophy of Logic Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic Model Theory in Logic and Philosophy of Logic Direct download (3 more) Export citation Bookmark 14 citations
Paires de structures o-minimales.Yerzhan Baisalov & Bruno Poizat - 1998 - Journal of Symbolic Logic 63 (2):570-578.details Mathematical Logic in Philosophy of Mathematics Direct download (9 more) Export citation Bookmark 15 citations
Paires De Structures O-Minimales.Yerzhan Baisalov & Bruno Poizat - 1998 - Journal of Symbolic Logic 63 (2):570-578.details Logic and Philosophy of Logic Export citation Bookmark 4 citations

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