Maximal small extensions of o-minimal structures

TL;DR

This paper constructs o-minimal structures with maximal small extensions, answering a question of Marker and demonstrating the existence of such structures across various cardinalities, including countable cases.

Contribution

It provides the first known examples of o-minimal structures with maximal small extensions for any cardinality, including the countable case.

Findings

Existence of o-minimal structures with maximal small extensions

Construction applicable to any cardinality

Maximal small extension can have maximal possible size in some cases

Abstract

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.