Elementary extensions of almost o-minimal structures

TL;DR

This paper proves that every almost o-minimal structure has a proper elementary extension that is also almost o-minimal, addressing a key question about the stability of this property under extensions.

Contribution

It establishes that almost o-minimality is preserved under elementary extensions, expanding understanding of the structure's model-theoretic properties.

Findings

Every almost o-minimal structure admits a proper elementary extension that is almost o-minimal.

Almost o-minimality is not preserved under elementary equivalence, but can be maintained through extensions.

The main theorem confirms the existence of such extensions for all almost o-minimal structures.

Abstract

This paper investigates almost o-minimal structures, a weakening of o-minimality introduced by Fujita to capture structures that lie outside the classical o-minimal framework. In contrast to o-minimality and local o-minimality, almost o-minimality is not preserved under elementary equivalence. This raises the natural question of whether every almost o-minimal structure admits a proper elementary extension that is again almost o-minimal. The main result of this paper provides an affirmative answer to this question.