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.     

Some definable properties of sets in non-valuational weakly o-minimal structures

Let \({\mathcal {M}}=(M,<,+,\cdot ,\ldots )\) be a non-valuational weakly o-minimal expansion of a real closed field \((M,<,+,\cdot )\). In this paper, we prove that \({\mathcal {M}}\) has a \(C^r\)-strong cell decomposition property, for each positive integer r, a best analogous result from Tanaka and Kawakami (Far East J Math Sci (FJMS) 25(3):417–431, 2007). We also show that curve selection property holds in non-valuational weakly o-minimal expansions of ordered groups. Finally, we extend the notion of definable compactness suitable for weakly o-minimal structures which was examined for definable sets (Peterzil and Steinhorn in J Lond Math Soc 295:769–786, 1999), and prove that a definable set is definably compact if and only if it is closed and bounded.     

Simple algebraic and semialgebraic groups over real closed fields

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

     

An o-minimal structure without mild parameterization

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.     

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.     

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.     

The Field of Reals with Multisummable Series and the Exponential Function

We show that the field of real numbers with multisummable realpower series is model complete, o-minimal and polynomially bounded.Further expansion by the exponential function yields again amodel complete and o-minimal structure which is exponentiallybounded, and in which the Gamma function on the positive realline is definable. 2000 Mathematics Subject Classification:primary 03C10, 32B05, 32B20; secondary, 26E05.     

Permutation Groups in o-Minimal Structures

In this paper we develop a structure theory for transitive permutationgroups definable in o-minimal structures. We fix an o-minimalstructure M, a group G definable in M, and a set and a faithfultransitive action of G on definable in M, and talk of the permutationgroup (G, ). Often, we are concerned with definably primitivepermutation groups (G, ); this means that there is no propernon-trivial definable G-invariant equivalence relation on ,so definable primitivity is equivalent to a point stabiliserG being a maximal definable subgroup of G. Of course, sinceany group definable in an o-minimal structure has the descendingchain condition on definable subgroups [23] we expect many questionson definable transitive permutation groups to reduce to questionson definably primitive ones. Recall that a group G definable in an o-minimal structure issaid to be connected if there is no proper definable subgroupof finite index. In some places, if G is a group definable inM we must distinguish between definability in the full ambientstructure M and G-definability, which means definability inthe pure group G:= (G, .); for example, G is G-definably connectedmeans that G does not contain proper subgroups of finite indexwhich are definable in the group structure. By definable, wealways mean definability in M. In some situations, when thereis a field R definable in M, we say a set is R-semialgebraic,meaning that it is definable in (R, +, .). We call a permutationgroup (G, ) R-semialgebraic if G, and the action of G on canall be defined in the pure field structure of a real closedfield R. If R is clear from the context, we also just write‘semialgebraic’.     

Invariance results for definable extensions of groups

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.     

Unstable structures definable in o-minimal theories

Let M{\mathcal {M}} be a dense o-minimal structure, N{\mathcal {N}} an unstable structure interpretable in M{\mathcal {M}}. Then there exists X, definable in N^eq{\mathcal {N}^{eq}}, such that X, with the induced N{\mathcal {N}}-structure, is linearly ordered and o-minimal with respect to that ordering. As a consequence we obtain a classification, along the lines of Zilber’s trichotomy, of unstable t-minimal types in structures interpretable in o-minimal theories.     

Vanishing homology

In this paper, we introduce a new homology theory devoted to the study of families such as semialgebraic or subanalytic families, and in general, to any family definable in an o-minimal structure (such as Denjoy–Carleman definable, or ln-exp definable sets). The idea is to study the cycles that are vanishing when we approach a special fiber. This also enables us to derive local metric invariants for germs of definable sets. We prove that the homology groups are finitely generated.     

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)     

Strongly minimal expansions of (C, +) definable in o-minimal fields

We characterize those functions f: definable in o-minimalexpansions of the reals for which the structure (,+, f) is stronglyminimal: such functions must be complex constructible, possiblyafter conjugating by a real matrix. In particular we prove aspecial case of the Zilber Dichotomy: an algebraically closedfield is definable in certain strongly minimal structures whichare definable in an o-minimal field.     

Weakly quasi-o-minimal models

We introduce the notion of a weakly quasi-o-minimal model and prove that such models lack the independence property. We show that every weakly quasi-o-minimal ordered group is Abelian, every divisible Archimedean weakly quasi-o-minimal ordered group is weakly o-minimal, and every weakly o-minimal quasi-o-minimal ordered group is o-minimal. We also prove that every weakly quasi-o-minimal Archimedean ordered ring with nonzero multiplication is a real closed field that is embeddable into the field of reals.     

Definable functions in the ℵ<Subscript>0</Subscript>-categorical weakly circularly minimal structures

We continue studying the analogs of o-minimality and weak o-minimality for circularly ordered sets. We present a complete characterization of the behavior of unary definable functions in an ?₀-categorical 1-transitive weakly circularly minimal structure. Using it, we describe the ?₀-categorical 1-transitive nonprimitive weakly circularly minimal structures of convexity rank greater than 1 up to binarity.     

Computing o-minimal topological invariants using differential topology

We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of the degree we obtain a new proof for the existence of torsion points in a definably compact group, and also a new proof of an o-minimal analogue of the Brouwer fixed point theorem.

     

Locally injective maps in O-minimal structures without poles are surjective

If is continuous and locally injective, then is in fact surjective and a homeomorphism, provided is definable in an o-minimal expansion without poles of the ordered additive group of real numbers; `without poles' means that every one-variable definable function is locally bounded. Some general properties of definable maps in o-minimal expansions of ordered abelian groups without poles are also established.

     

Tame properties of sets and functions definable in weakly o-minimal structures

Let ${{\mathcal{M}}=(M, <, \ldots )}$ be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in ${{\mathcal{M}}}$ 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 ${{\mathcal{M}}}$ satisfy an extended version of IVP. After introducing a weak version of definable connectedness in ${{\mathcal{M}}}$ , we prove that strong cells in ${{\mathcal{M}}}$ are weakly definably connected, so every set definable in ${{\mathcal{M}}}$ is a finite union of its weakly definably connected components, provided that ${{\mathcal{M}}}$ has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in ${{\mathcal{M}}}$ and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph (NDG) which was examined for definable functions in (Dolich et al. in Trans. Am. Math. Soc. 362:1371–1411, 2010) and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure ${{\mathcal{M}}}$ having cell decomposition, satisfies NDG, i.e. every definable function in ${{\mathcal{M}}}$ has no dense graph.     

Compact domination for groups definable in linear o-minimal structures

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.     

Compression Schemes, Stable Definable Families, and o-Minimal Structures

We show that any family of sets uniformly definable in an o-minimal structure has an extended compression scheme of size equal to the number of parameters in the defining formula.