A box type is an n-type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of a polynomially bounded o-minimal structure M. From this, we deduce various structure theorems for subsets of $M^k $ , definable in the expansion M of M by all convex subsets of the line. We show that M after naming constants, is model complete provided M is model complete.

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.

We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve has a definable extension to a closed set. This situation is translated into a question about types: What are the conditions on an n-type such that, for any bounded definable function, the germ of the function on the type has a definable continuous global extension? Certain categories of (...) definable types have this property, and we give the precise conditions that are equivalent to existence of the global extension. (shrink)

Given an 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} with a group operation, we show that for a properly convex subset U, the theory of the expanded 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} has definable Skolem functions precisely when 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 valuational. As a corollary, we get an elementary proof that the theory of any such 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} does not satisfy definable choice. (shrink)

We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.

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.