Dirichlet regularity in arbitrary o‐minimal structures on the field ℝ up to dimension 4

In this article we show that the set of Dirichlet regular boundary points of a bounded domain of dimension up to 4, definable in an arbitrary o‐minimal structure on the field ?, is definable in the same structure. Moreover we give estimates for the dimension of the set of non‐regular boundary points, depending on whether the structure is polynomially bounded or not. This paper extends the results from the author's Ph.D. thesis [6, 7] where the problem was solved for polynomially bounded o‐minimal structures expanding the real field. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Filling certain cuts in discrete weakly o‐minimal structures

Discrete weakly o‐minimal structures, although not so stimulating as their dense counterparts, do exhibit a certain wealth of examples and pathologies. For instance they lack prime models and monotonicity for definable functions, and are not preserved by elementary equivalence. First we exhibit these features. Then we consider a countable theory of weakly o‐minimal structures with infinite definable discrete (convex) subsets and we study the Boolean algebra of definable sets of its countable models. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Definability and nondefinability results for certain o‐minimal structures

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] → ? there is a definable C^∞ function g: [–ε, 1] → ?, for some ε > 0, such that g (x) = g₀(x) for all x ∈ [0, 1] (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 strong cell decomposition property for weakly o‐minimal structures

We consider a class of weakly o‐minimal structures admitting an o‐minimal style cell decomposition, for which one can construct certain canonical o‐minimal extension. The paper contains several fundamental facts concerning the structures in question. Among other things, it is proved that the strong cell decomposition property is preserved under elementary equivalences. We also investigate fiberwise properties (of definable sets and definable functions), definable equivalence relations, and conditions implying elimination of imaginaries.     

Singularities of o‐minimal Peano derivatives

The notion of Peano differentiability generalizes the differentiability in the usual sense to higher order. Peano differentiable functions have derivatives which are sometimes differentiable or continuous or not even locally bounded. We give a complete characterisation of the sets in which Peano differentiable functions which are definable in an o‐minimal expansion of a real closed field are continuously differentiable. Thereby, we also distinguish between several kinds of discontinuities (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

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)     

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)     

Uniform bounds on growth in o‐minimal structures

We prove that a function definable with parameters in an o‐minimal structure is bounded away from ∞ as its argument goes to ∞ by a function definable without parameters, and that this new function can be chosen independently of the parameters in the original function. This generalizes a result in [1]. Moreover, this remains true if the argument is taken to approach any element of the structure (or ±∞), and the function has limit any element of the structure (or ±∞) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Expansions of ordered fields without definable gaps

In this paper we are concerned with definably, with or without parameters, (Dedekind) complete expansions of ordered fields, i. e. those with no definable gaps. We present several axiomatizations, like being definably connected, in each of the two cases. As a corollary, when parameters are allowed, expansions of ordered fields are o‐minimal if and only if all their definable subsets are finite disjoint unions of definably connected (definable) subsets. We pay attention to how simply (in terms of the quantifier complexity and/or usage of parameters) a definable gap in an expansion is so. Next we prove that over parametrically definably complete expansions of ordered fields, all one‐to‐one definable (with parameters) continuous functions are monotone and open. Moreover, in both parameter and parameter‐free cases again, definably complete expansions of ordered fields satisfy definable versions of the Heine‐Borel and Extreme Value theorems and also Bounded Intersection Property for definable families of closed bounded subsets.     

Minimality conditions on circularly ordered structures

We explore analogues of o‐minimality and weak o‐minimality for circularly ordered sets. Much of the theory goes through almost unchanged, since over a parameter the circular order yields a definable linear order. Working over ?? there are differences. Our main result is a structure theory (with infinitely many doubly transitive examples related to Jordan permutation groups) for ?₀‐categorical weakly circularly minimal structures. There is a 5‐homogeneous (or ‘5‐indiscernible’) example which is not 6‐homogeneous, but any example which is k‐homogeneous for some k ≥ 6 is k‐homogeneous for all k. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weakly o‐Minimal Expansions of Boolean Algebras

We propose a definition of weak o‐minimality for structures expanding a Boolean algebra. We study this notion, in particular we show that there exist weakly o‐minimal non o‐minimal examples in this setting.     

ω‐categorical weakly o‐minimal expansions of Boolean lattices

We study ω‐categorical weakly o‐minimal expansions of Boolean lattices. We show that a structure ?? = (A,≤, ?) expanding a Boolean lattice (A,≤) by a finite sequence I of ideals of A closed under the usual Heyting algebra operations is weakly o‐minimal if and only if it is ω‐categorical, and hence if and only if A/I has only finitely many atoms for every I ∈ ?. We propose other related examples of weakly o‐minimal ω‐categorical models in this framework, and we examine the internal structure of these models.     

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.     

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)     

A note on stable sets,groups, and theories with NIP

Let M be an arbitrary structure. Then we say that an M ‐formula φ (x) defines a stable set in M if every formula φ (x) ∧ α (x, y) is stable. We prove: If G is an M ‐definable group and every definable stable subset of G has U ‐rank at most n (the same n for all sets), then G has a maximal connected stable normal subgroup H such that G /H is purely unstable. The assumptions hold for example if M is interpretable in an o‐minimal structure. More generally, an M ‐definable set X is weakly stable if the M ‐induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Measurable groups of low dimension

We consider low‐dimensional groups and group‐actions that are definable in a supersimple theory of finite rank. We show that any rank 1 unimodular group is (finite‐by‐Abelian)‐by‐finite, and that any 2‐dimensional asymptotic group is soluble‐by‐finite. We obtain a field‐interpretation theorem for certain measurable groups, and give an analysis of minimal normal subgroups and socles in groups definable in a supersimple theory of finite rank where infinity is definable. We prove a primitivity theorem for measurable group actions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximal small extensions of o‐minimal structures

A proper elementary extension of a model is called small if it realizes no new types over any finite set in the base model. We answer a question of Marker, and show that it is possible to have an o‐minimal structure with a maximal small extension. Our construction yields such a structure for any cardinality. We show that in some cases, notably when the base structure is countable, the maximal small extension has maximal possible cardinality (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)