On the Euler characteristic of definable groups

We show that in an arbitrary o‐minimal structure the following are equivalent: (i) conjugates of a definable subgroup of a definably connected, definably compact definable group cover the group if the o‐minimal Euler characteristic of the quotient is non zero; (ii) every infinite, definably connected, definably compact definable group has a non trivial torsion point (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

on the o-minimal LS-category

We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LS-category of definable groups. Along the way, we show that two definably connected definably compact definable groups G and H are definable homotopy equivalent if and only if L(G) and L(H) are homotopy equivalent, where L is the functor which associates to each definable group its corresponding Lie group via Pillay’s conjecture.     

A remark on divisibility of definable groups

We show that if G is a definably compact, definably connected definable group defined in an arbitrary o‐minimal structure, then G is divisible. Furthermore, if G is defined in an o‐minimal expansion of a field, k ∈ ? and p_k : G → G is the definable map given by p_k (x ) = x^k for all x ∈ G , then we have |(p_k )^–1(x )| ≥ k^r for all x ∈ G , where r > 0 is the maximal dimension of abelian definable subgroups of G . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Noetherian varieties in definably complete structures

We prove that the zero-set of a C ^∞ function belonging to a noetherian differential ring M can be written as a finite union of C ^∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zero-dimensional regular zero-set of functions in M consists of finitely many points. These results hold not only for C ^∞ functions over the reals, but more generally for definable C ^∞ functions in a definably complete expansion of an ordered field. The class of definably complete expansions of ordered fields, whose basic properties are discussed in this paper, expands the class of real closed fields and includes o-minimal expansions of ordered fields. Finally, we provide examples of noetherian differential rings of C ^∞ functions over the reals, containing non-analytic functions.     

Pseudo definably connected definable sets

In o‐minimal structures, every cell is definably connected and every definable set is a finite union of its definably connected components. In this note, we introduce pseudo definably connected definable sets in weakly o‐minimal structures having strong cell decomposition, and prove that every strong cell in those structures is pseudo definably connected. It follows that every definable set can be written as a finite union of its pseudo definably connected components. We also show that the projections of pseudo definably connected definable sets are pseudo definably connected. Finally, we compare pseudo definable connectedness with (recently introduced) weak definable connectedness of definable sets in weakly o‐minimal structures.     

The universal covering homomorphism in o‐minimal expansions of groups

Suppose G is a definably connected, definable group in an o‐minimal expansion of an ordered group. We show that the o‐minimal universal covering homomorphism : → G is a locally definable covering homomorphism and π₁(G) is isomorphic to the o‐minimal fundamental group π (G) of G defined using locally definable covering homomorphisms. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decomposition into special submanifolds

We study definably complete locally o-minimal expansions of ordered groups. We propose a notion of special submanifolds with tubular neighborhoods and show that any definable set is decomposed into finitely many special submanifolds with tubular neighborhoods.     

Uniformly locally o-minimal structures and locally o-minimal structures admitting local definable cell decomposition

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.     

Fundamental group in o-minimal structures with definable Skolem functions

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.     

Definability of types and VC density in differential topological fields

Given a model-complete theory of topological fields, we considered its generic differential expansions and under a certain hypothesis of largeness, we axiomatised the class of existentially closed ones. Here we show that a density result for definable types over definably closed subsets in such differential topological fields. Then we show two transfer results, one on the VC-density and the other one, on the combinatorial property NTP2.     

Nonstandard models that are definable in models of Peano Arithmetic

In this paper, we investigate definable models of Peano Arithmetic PA in a model of PA. For any definable model N without parameters in a model M, we show that N is isomorphic to M if M is elementary extension of the standard model and N is elementarily equivalent to M. On the other hand, we show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M but N is not isomorphic to M. We also show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M, and N is isomorphic to M, but N is not definably isomorphic to M. And also, we give a generalization of Tennenbaum's theorem. At the end, we give a new method to construct a definable model by a refinement of Kotlarski's method. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Definable Compactness and Definable Subgroups of o-Minimal Groups

The paper introduces the notion of definable compactness andwithin the context of o-minimal structures proves several topologicalproperties of definably compact spaces. In particular a definableset in an o-minimal structure is definably compact (with respectto the subspace topology) if and only if it is closed and bounded.Definable compactness is then applied to the study of groupsand rings in o-minimal structures. The main result proved isthat any infinite definable group in an o-minimal structurethat is not definably compact contains a definable torsion-freesubgroup of dimension 1. With this theorem, a complete characterizationis given of all rings without zero divisors that are definablein o-minimal structures. The paper concludes with several examplesillustrating some limitations on extending the theorem.     

On the Boolean algebras of definable sets in weakly o‐minimal theories

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 (hence T is p‐ω‐categorical), 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 (convex) 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. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Definable group extensions in semi‐bounded o‐minimal structures

In this note we show: Let R = 〈R, <, +, 0, …〉 be a semi‐bounded (respectively, linear) o‐minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈R^m, +〉 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

First-order definability and algebraicity of the sets of annihilating and generating collections of elements for some relatively free solvable groups

We study the first-order definable, Diophantine, and algebraic subsets in the set of all ordered sets generating a group or generating a group as a normal subgroup for some relatively free solvable groups.     

Smooth functions in o-minimal structures

Fix an o-minimal expansion of the real exponential field that admits smooth cell decomposition. We study the density of definable smooth functions in the definable continuously differentiable functions with respect to the definable version of the Whitney topology. This implies that abstract definable smooth manifolds are affine. Moreover, abstract definable smooth manifolds are definably C^∞-diffeomorphic if and only if they are definably C¹-diffeomorphic.     

Groups,group actions and fields definable in first‐order topological structures

Given a group (G, ·), G?M^m, definable in a first‐order structure $\mathcal {M}=(M,\ldots )Given a group (G, ·), G?M^m, definable in a first‐order structure $\mathcal {M}=(M,\ldots )$ equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V?G and define a new topology τ on G with which (G, ·) becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from M^m. Likewise we topologize transitive group actions and fields definable in $\mathcal {M}$. These results require a series of preparatory facts concerning dimension functions, some of which might be of independent interest.     

Notes on local o‐minimality

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 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The ordering of commutative terms

By a commutative term we mean an element of the free commutative groupoid F of infinite rank. For two commutative terms a, b write a ⩽ b if b contains a subterm that is a substitution instance of a. With respect to this relation, F is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other things, we prove that every commutative term (or its block in the factor) is a definable element. Consequently, the ordered set has no automorphisms except the identity.     

Definable subgroups of measure algebras

We show that type‐definable subgroups of measure algebras are definable. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)