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  1. The field of reals with a predicate for the real algebraic numbers and a predicate for the integer powers of two.Mohsen Khani - 2015 - Archive for Mathematical Logic 54 (7-8):885-898.
    Given a theory T of a polynomially bounded o-minimal expansion R of R¯=⟨R,+,.,0,1,<⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\mathbb{R}} = \langle\mathbb{R}, +,., 0, 1, < \rangle}$$\end{document} with field of exponents Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}$$\end{document}, we introduce a theory T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{T}}$$\end{document} whose models are expansions of dense pairs of models of T by a discrete multiplicative group. We prove that T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...)
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  • On non-compact p-adic definable groups.Will Johnson & Ningyuan Yao - 2022 - Journal of Symbolic Logic 87 (1):188-213.
    In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is not definably compact, then G contains a definable torsion-free subgroup of dimension 1. We prove here a p-adic analogue of the Peterzil–Steinhorn theorem, in the special case of abelian groups. Let G be an abelian group definable in a p-adically closed field M. If G is not definably compact then there is a definable subgroup H of dimension 1 which is not definably compact. (...)
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  • Wild theories with o-minimal open core.Philipp Hieronymi, Travis Nell & Erik Walsberg - 2018 - Annals of Pure and Applied Logic 169 (2):146-163.
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  • Distality for the Asymptotic Couple of the Field of Logarithmic Transseries.Allen Gehret & Elliot Kaplan - 2020 - Notre Dame Journal of Formal Logic 61 (2):341-361.
    We show that the theory Tlog of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP, this provides a new proof that Tlog is NIP.
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  • Non-standard lattices and o-minimal groups.Pantelis E. Eleftheriou - 2013 - Bulletin of Symbolic Logic 19 (1):56-76.
    We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a lattice. We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry and (...)
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  • NIP for some pair-like theories.Gareth Boxall - 2011 - Archive for Mathematical Logic 50 (3-4):353-359.
    Generalising work of Berenstein, Dolich and Onshuus (Preprint 145 on MODNET Preprint server, 2008) and Günaydın and Hieronymi (Preprint 146 on MODNET Preprint server, 2010), we give sufficient conditions for a theory TP to inherit N I P from T, where TP is an expansion of the theory T by a unary predicate P. We apply our result to theories, studied by Belegradek and Zilber (J. Lond. Math. Soc. 78:563–579, 2008), of the real field with a subgroup of the unit (...)
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  • Fields with a dense-codense linearly independent multiplicative subgroup.Alexander Berenstein & Evgueni Vassiliev - 2020 - Archive for Mathematical Logic 59 (1-2):197-228.
    We study expansions of an algebraically closed field K or a real closed field R with a linearly independent subgroup G of the multiplicative group of the field or the unit circle group \\), satisfying a density/codensity condition. Since the set G is neither algebraically closed nor algebraically independent, the expansion can be viewed as “intermediate” between the two other types of dense/codense expansions of geometric theories: lovely pairs and H-structures. We show that in both the algebraically closed field and (...)
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