Many scholars and activists seek to eliminate “race”—the word and the concept—from our vocabulary. Their claim is clear: because science has shown that racial essentialism is false and because the idea of race has proved virulent, we should do away with the concept entirely. Michael O. Hardimon criticizes this line of thinking, arguing that we must recognize the real ways in which race exists in order to revise our understanding of its significance. Rethinking Race provides a novel answer to the (...) question “What is race?” Pernicious, traditional racialism maintains that people can be judged and ranked according to innate racial features. Hardimon points out that those who would eliminate race make the mistake of associating the word only with this view. He agrees that this concept should be jettisoned, but draws a distinction with three alternative ideas: first, a stripped-down version of the ordinary concept of race that recognizes minimal physical differences between races but does not consider them significant; second, a scientific understanding of populations with shared lines of descent; and third, an acknowledgment of “socialrace” as a separate construction. Hardimon provides a language for understanding the ways in which races do and do not exist. His account is realistic in recognizing the physical features of races, as well as the existence of races in our social world. But it is deflationary in rejecting the concept of hierarchical or defining racial characteristics. Ultimately, Rethinking Race offers a philosophical basis for repudiating racism without blinding ourselves to reality. (shrink)

A totally ordered group G is essentially periodic if for every definable non-trivial convex subgroup H of G every definable subset of G is equal to a finite union of cosets of subgroups of G on some interval containing an end segment of H; it is coset-minimal if all definable subsets are equal to a finite union of cosets, intersected with intervals. We study definable sets and functions in such groups, and relate them to the quasi-o-minimal groups introduced in Belegradek (...) et al. . Main results: An essentially periodic group G is abelian; if G is discrete, then definable functions in one variable are ultimately piecewise linear. A group such that every model elementarily equivalent to it is coset-minimal is quasi-o-minimal , and its definable functions in one variable are piecewise linear. (shrink)

We use a result due to Rolin, Speissegger, and Wilkie to show that definable sets in certain o-minimal structures admit definable parameterizations by mild maps. We then use this parameterization to prove a result on the density of rational points on curves defined by restricted Pfaffian functions.

We show that in an o-minimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in o-minimal expansions of the ordered set of real numbers.

The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$, with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$. We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$, and (...) the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$, so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$. This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case. (shrink)

A totally ordered group G is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G{±∞}. Continuing work in Belegradek et al. 1115) and Point and Wagner 261), we study coset-minimality, as well as two weak versions of the notion: eventual and ultimate coset-minimality. These groups are abelian; an eventually coset-minimal group, as a pure ordered group, is an ordered abelian group of finite regular rank. Any (...) pure ordered abelian group of finite regular rank is ultimately coset-minimal and has the exchange property; moreover, every definable function in such a group is piecewise linear. Pure coset-minimal and eventually coset-minimal groups are classified. In a discrete coset-minimal group every definable unary function is piece-wise linear 261), where coset-minimality of the theory of the group was required). A dense coset-minimal group has the exchange property ); moreover, any definable unary function is piecewise linear, except possibly for finitely many cosets of the smallest definable convex nonzero subgroup. Finally, we give some examples and open questions. (shrink)

We consider the extent to which certain properties of definably complete structures may persist in structures which are not definably complete, particularly in the weakly o-minimal structures.

We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic properties when the given structure is a locally o-minimal structure which (...) admits local definable cell decomposition. (shrink)

To characterize human emotions, researchers have increasingly utilized Automatic Facial Expression Analysis (AFEA), which automates the Facial Action Coding System (FACS) and translates the facial muscular positioning into the basic universal emotions. There is broad interest in the application of FACS for assessing consumer expressions as an indication of emotions to consumer product-stimuli. However, the translation of FACS to characterization of emotions is elusive in the literature. The aim of this systematic review is to give an overview of how FACS (...) has been used to investigate human emotional behavior to consumer product-based stimuli. The search was limited to studies published in English after 1978, conducted on humans, using FACS or its action units to investigate affect, where emotional response is elicited by consumer product-based stimuli evoking at least one of the five senses. The search resulted in an initial total of 1,935 records, of which 55 studies were extracted and categorized based on the outcomes of interest including (i) method of FACS implementation; (ii) purpose of study; (iii) consumer product-based stimuli used; and (iv) measures of affect validation. Most studies implemented FACS manually (73%) to develop products and/or software (20%) and used consumer product-based stimuli that had known and/or defined capacity to evoke a particular affective response, such as films and/or movie clips (20%); minimal attention was paid to consumer products with low levels of emotional competence or with unknown affective impact. The vast majority of studies (53%) did not validate FACS-determined affect and, of the validation measures that were used, most tended to be discontinuous in nature and only captured affect as it holistically related to an experience. This review illuminated some inconsistencies in how FACS is carried out as well as how emotional response is inferred from facial muscle activation. This may prompt researchers to consider measuring the total consumer experience by employing a variety of methodologies in addition to FACS and its emotion-based interpretation guide. Such strategies may better conceptualize consumers' experience with products of low, unknown, and/or undefined capacity to evoke an affective response such as product prototypes, line extensions, etc. (shrink)

Let $\scr{M}=\langle M,+,<,0,S\rangle $ be a linear o-minimal expansion of an ordered group, and $G=\langle G,\oplus ,e_{G}\rangle $ an n-dimensional group definable in $\scr{M}$ . We show that if G is definably connected with respect to the t-topology, then it is definably isomorphic to a definable quotient group U/L, for some convex ${\ssf V}\text{-definable}$ subgroup U of $\langle M^{n},+\rangle $ and a lattice L of rank equal to the dimension of the 'compact part' of G.

In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allow us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally (...) definable covering maps; monodromy equivalence for locally constant o-minimal sheaves – from which one obtains, as in algebraic topology, classification results for locally definable covering maps, o-minimal Hurewicz and Seifert–van Kampen theorems. (shrink)

It is shown that if is an o-minimal structure such that is a dense total order and ≾ is a parameter-definable partial order on M, then ≾ has an extension to a definable total order.

We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. A decomposition theorem into quasi-special submanifolds is also demonstrated.

A structure (M, $ ,...) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal (...) ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1. (shrink)

We prove the Compact Domination Conjecture for groups definable in linear o-minimal structures. Namely, we show that every definably compact group G definable in a saturated linear o-minimal expansion of an ordered group is compactly dominated by (G/G 00, m, π), where m is the Haar measure on G/G 00 and π : G → G/G 00 is the canonical group homomorphism.

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 (...) real closed fields. Here we further develop this analogy. We establish an o-minimal style cell decomposition for weakly o-minimal non-valuational expansions of ordered groups. For structures enjoying such a strong cell decomposition we construct a canonical o-minimal extension. Finally, we make attempts towards generalizing the o-minimal Euler characteristic to the class of sets definable in weakly o-minimal structures with the strong cell decomposition property. (shrink)

By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal "infinitesimal subgroup" G00 such that the quotient G/G00, equipped with the "logic topology", is a compact (real) Lie group. Our first result is that the functor G → G/G00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum (...) G̃ of G. We prove that G/G00 is a topological quotient of G̃. We thus obtain a natural homomorphism ψ* from the cohomology of G/G00 to the (Čech-)cohomology of G̃. We show that if G00 satisfies a suitable contractibility conjecture then $\widetilde{G^{00}}$ is acyclic in Čech cohomology and ψ* is an isomorphism. Finally we prove the conjecture in some special cases. (shrink)

Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let $f:X \rightarrow R^n$ be a definable map, where X is a definable set and R is the universe of the structure. We demonstrate the inequality $\dim ) \leq \dim $ in this paper. As a corollary, we get that the set of the points at which f is discontinuous is of dimension smaller than $\dim $. We also show that the (...) structure is definably Baire in the course of the proof of the inequality. (shrink)

The topology of definable sets in an o-minimal expansion of a group is not fully understood due to the lack of a triangulation theorem. Despite the general validity of the cell decomposition theorem, we do not know whether any definably compact set is a definable CW-complex. Moreover the closure of an o-minimal cell can have arbitrarily high Betti numbers. Nevertheless we prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated (...) and invariant under elementary extensions and expansions of the language. (shrink)

In this note we show: Let R = 〈R, <, +, 0, …〉 be a semi-bounded o-minimal expansion of an ordered group, and G a group definable in R of linear dimension m . Then G is a definable extension of a bounded definable group B by 〈Rm, +〉.

We study first-order expansions of ordered fields that are definably complete, and moreover either are locally o-minimal, or have a locally o-minimal open core. We give a characterisation of structures with locally o-minimal open core, and we show that dense elementary pairs of locally o-minimal structures have locally o-minimal open core.

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o‐minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good‐shaped definable subsets called quasi‐special submanifolds. Using this fact, we investigate definably complete locally o‐minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o‐minimal (...) expansions of ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo‐definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo‐definable spaces are definably compact. Here, a pseudo‐definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set‐valued functions with definably compact domains of definition. (shrink)

We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures (...) in the special cases when G is 1-dimensional, and when G is definably simple. (shrink)

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets (...) of Cartesian products of definable sets, showing that if X, Y and S are non-empty definable sets and S is a large subset of X × Y, then for a large set of tuples $\langle \overline{a}_{1},\ldots,\overline{a}_{2^{k}}\rangle \in X^{2^{k}}$ , where k = dim(Y), the union of fibers $S_{\overline{a}_{1}}\cup \cdots \cup S_{\overline{a}_{2^{k}}}$ is large in Y. Finally, given a weakly o-minimal structure ������, we find various conditions equivalent to the fact that the topological dimension in ������ enjoys the addition property. (shrink)

The main goal of this note is to study for certain o-minimal structures the following propriety: for each definable C∞ function g0: [0, 1] → ℝ there is a definable C∞ function g: [–ε, 1] → ℝ, for some ε > 0, such that g = g0 for all x ∈ [0, 1].

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology theory for abstract-definable $C^{\infty (...) }$ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer–Vietoris sequence and the invariance under abstract-definable $C^{\infty }$ diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must work in a tame context that defines sufficiently many primitives and assume the validity of a statement related to Bröcker’s question.Abstract prepared by Rodrigo Figueiredo.E-mail: [email protected]: https://doi.org/10.11606/T.45.2019.tde-28042019-181150. (shrink)

An argument of A. Borel [Bor—61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

We consider the sets definable in the countable models of a weakly o-minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic , in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within (...) expansions of Boolean lattices, every weakly o-minimal theory is p-ω-categorical. (shrink)

We investigate the group of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure . We prove the existence of a “large”, definable subgroup of . If contains an infinite definable set of homomorphisms then some definable subgroup of B admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure but also in any structure definable in.

We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest type-definable subgroup G00 of bounded index and G/G00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author.

We study o-minimal expansions of Archimedean totally ordered groups. We first prove that any such expansion must be elementarily embeddable via a unique (provided some nonzero element is 0-definable) elementary embedding into a unique o-minimal expansion of the additive ordered group of real numbers R. We then show that a definable function in an o-minimal expansion of R enjoys good differentiability properties and use this to prove that an Archimedean real closed field is definable in any nonsemilinear expansion of R. (...) Combining these results, we obtain several restrictions on possible o-minimal expansions of arbitrary Archimedean ordered groups and in particular of the rational ordered group. (shrink)

Let 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} be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in 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} which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in 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} satisfy an extended (...) version of IVP. After introducing a weak version of definable connectedness in 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}, we prove that strong cells in 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} are weakly definably connected, so every set definable in 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 a finite union of its weakly definably connected components, provided that 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 the strong cell decomposition property. Then, we consider a local continuity property for definable functions in 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} and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph which was examined for definable functions in and related to uniform finiteness, definable completeness, and others. We show that every weakly 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} having cell decomposition, satisfies NDG, i.e. every definable function in 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 no dense graph. (shrink)

Let be an o-minimal structure expanding a real closed field R. We show that any definable set in Rn can be stratified into cells, whose defining functions are smooth Lipschitz continuous functions with constant 2n3/2, which have additional regularity conditions on the derivatives of higher order.

Let (M, ≤,...) denote a Boolean ordered o-minimal structure. We prove that a Boolean subalgebra of M determined by an algebraically closed subset contains no dense atoms. We show that Boolean algebras with finitely many atoms do not admit proper expansions with o-minimal theory. The proof involves decomposition of any definable set into finitely many pairwise disjoint cells, i.e., definable sets of an especially simple nature. This leads to the conclusion that Boolean ordered structures with o-minimal theories are essentially bidefinable (...) with Boolean algebras with finitely many atoms, expanded by naming constants. We also discuss the problem of existence of proper o-minimal expansions of Boolean algebras. (shrink)

We prove, by explicit construction, that not all sets definable in polynomially bounded o-minimal structures have mild parameterization. Our methods do not depend on the bounds particular to the definition of mildness and therefore our construction is also valid for a generalized form of parameterization, which we call G-mild. Moreover, we present a cell decomposition result for certain o-minimal structures which may be of independent interest. This allows us to show how our construction can produce polynomially bounded, model complete expansions (...) of the real ordered field which, in addition to lacking G-mild parameterization, nonetheless still have analytic cell decomposition. (shrink)

Let T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L * be L together with a unary predicate P. Let T * be the L * -theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an |M| + -saturated elementary extension of N (and (...) M is the interpretation of P). Using the definability of types result, we show that T * is complete and we give a simple set of axioms for T * . We also show that for every L * -formula φ(x) there is an L-formula ψ(x) such that $T^\ast \models (\forall \mathbf{x})(P(\mathbf{x}) \rightarrow (\phi(\mathbf{x}) \mapsto \psi (\mathbf{x}))$ . This yields the following result: Let M be a Dedekind complete model of T. Let φ(x, y) be an L-formula where l(y) = k. Let $\mathbf{X} = \{X \subset M^k$ : for some a in an elementary extension N of M, X = φ (a,y N ∩ M k }. Then there is a formula ψ(y, z) of L such that X = {ψ (y, b) M : b in M}. (shrink)

Here we work in an arbitrary o-minimal expansion of a divisible ordered abelian group. We say that a definable ring is definably semiprime if squares of nontrivial two-sided ideals definable in the expansion are nontrivial. We prove a definable version of Wedderburn–Artin theorem and give a characterization of definably semiprime rings.

In this paper we study the metric spaces that are definable in a polynomially bounded o-minimal structure. We prove that the family of metric spaces definable in a given polynomially bounded o-minimal structure is characterized by the valuation field Λ of the structure. In the last section we prove that the cardinality of this family is that of Λ. In particular these two results answer a conjecture given in [SS] about the countability of the metric types of analytic germs. The (...) proof is a mixture of geometry and model theory. (shrink)

Let P be the ω-orbit of a point under a unary function definable in an o-minimal expansion ℜ of a densely ordered group. If P is monotonically cofinal in the group, and the compositional iterates of the function are cofinal at +\infty in the unary functions definable in ℜ, then the expansion (ℜ, P) has a number of good properties, in particular, every unary set definable in any elementarily equivalent structure is a disjoint union of open intervals and finitely many (...) discrete sets. (shrink)

We introduce and study some local versions of o-minimality, requiring that every definable set decomposes as the union of finitely many isolated points and intervals in a suitable neighbourhood of every point. Motivating examples are the expansions of the ordered reals by sine, cosine and other periodic functions.

Let M be a big o-minimal structure and G a type-definable group in Mⁿ. We show that G is a type-definable subset of a definable manifold in Mⁿ that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some Mk with the topology induced by Mk. Part of this result holds for the wider class of so-called invariant (...) groups: each invariant group G in Mⁿ has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as Mⁿ. (shrink)

We study the Hardy field associated with an o-minimal expansion of the real numbers. If the set of analytic germs is dense in the Hardy field, then we can definably analytically separate sets in , and we can definably analytically approximate definable continuous unary functions. A similar statement holds for definable smooth functions.

The following two assertions are equivalent for an o-minimal expansion of an ordered group $$\mathcal M=(M,<,+,0,\ldots )$$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function $$f:A \rightarrow M$$ defined on a definable closed subset of $$M^n$$ has a definable continuous extension $$F:M^n \rightarrow M$$.

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 (...) generic elements is each time introduced. (shrink)

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space \ \) is definably homeomorphic to an affine definable space with the induced subspace topology). One of the main results says that it is sufficient for X to be regular and decompose into finitely many definably connected components.

We prove weak elimination of imaginary elements for Boolean orderings with finitely many atoms. As a consequence we obtain equivalence of the two notions of o-minimality for Boolean ordered structures, introduced by C. Toffalori. We investigate atoms in Boolean algebras induced by algebraically closed subsets of Boolean ordered structures. We prove uniqueness of prime models in strongly o-minimal theories of Boolean ordered structures.