Simple product and locally o-minimal theories

TL;DR

This paper constructs examples of theories that are locally o-minimal and satisfy NIP but are not o-minimal, using the concept of simple product, expanding the landscape of model-theoretic classifications.

Contribution

It introduces the notion of simple product to construct theories with specific properties, expanding understanding of locally o-minimal and NIP theories.

Findings

Existence of theories that are locally o-minimal but not o-minimal.

Construction of such theories using simple product.

Examples include expansions of the real ordered group.

Abstract

There exist NIP and non-NTP$_2$ theories satisfying all the following conditions: It is not o-minimal; All models are strongly locally o-minimal; It has a model which is an expansion of the linearly ordered abelian group over the reals $(\mathbb R\;;\;0,+,<)$. We construct these examples using the notion of simple product.