Representing binary ordering relations by numerical functions is a basic problem of the theory of measurement. It has applications in many fields of science, and arises in psychology and economics as the problem of establishing utility functions for preference relations. We obtain definable utility representations for (both continuous and upper semicontinuous) definable preferences in o-minimal...

Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures. ∗With partial support from the FCT (Fundação para a Ciência e Tecnologia), program POCTI (Portugal/FEDER-EU) and Fundação Calouste Gulbenkian. †Supported by an EPSRC studentship. ‡Supported by a postdoctoral fellowship from CMAF, Universidade de Lisboa. MSC: 03C64; 55N30.

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 s...

In 1934, Whitney gave a necessary and sufficient condition on a jet of order m on a closed subset E of R to be the jet of order m of a C-function; jets satisfying this condition are known as C-Whitney fields. Later, Paw lucki and Kurdyka proved that subanalytic C-Whitney fields are jets of order m of sybanalytic C-functions. Here, we work in an o-minimal expansion of a real closed field and pro...

We work over an o-minimal expansion of a real closed field R. Given a closed simplicial complex K and some definable subsets S1, . . . , Sk of its realization |K| in R we prove that there exists a triangulation (K , φ) of |K| compatible with S1, . . . , Sk such thatK ′ is a subdivision of K and φ is definably homotopic to id|K|.

In [Berarducci-Otero, 2004], the authors define an analogue of Lebesgue measure for bounded definable subsets of an o-minimal structure K , which expands a real closed field. Our aim is defining the area of a bounded definable d-dimensional subsets of Kn. Different possible definitions for such area are possible: we will prove that some of them are equivalent. On the reals, there are many diffe...

Let M be a C-minimal structure and T its canonical tree (which corresponds in an ultrametric space to the set of closed balls with radius different than ∞ ordered by inclusion). We present a description of definable locally constant functions f : M → T in C-minimal structures having a canonical tree with infinitely many branches at each node and densely ordered branches. This provides both a de...

Given a polynomially bounded o–minimal structure S and a set A ⊂ R belonging to S, we show that A (considered as a subset of C) is pluriregular at every point a ∈ intA that can be attained by a C∞ arc γ : [0, ] → R belonging to S, such that γ(0) = a and γ((0, ]) ⊂ intA. In particular, if S is a recently found in [22] polynomially bounded o–minimal structure of quasianalytic functions in the sen...

The first papers on o-minimal structures appeared in the mid 1980s, since then the subject has grown into a wide ranging gener-alisation of semialgebraic, subanalytic and subpfaffian geometry. In these notes we will try to show that this is in fact the case by presenting several examples of o-minimal structures and by listing some geometric properties of sets and maps definable in o-minimal str...