Elementary extensions of almost o-minimal structures

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.

💡 Research Summary

The paper addresses a natural question raised by Fujita concerning the stability of the almost‑o‑minimal property under elementary extensions. An almost‑o‑minimal structure is a weakening of o‑minimality: every bounded definable set is a finite union of points and open intervals, while no global finiteness condition is imposed. This property is known to be preserved under taking elementary substructures but not under elementary equivalence; for example, the structure (ℚ,<,ℤ) is almost‑o‑minimal, yet a two‑copy lexicographic expansion of it is elementarily equivalent but fails the almost‑o‑minimal condition because a bounded interval contains infinitely many copies of ℤ. Consequently, it is not obvious whether every almost‑o‑minimal structure can be extended to a proper elementary superstructure that remains almost‑o‑minimal.

The authors answer this question affirmatively. Their proof combines two main technical tools: bounded ultrapowers and a variant of Gaifman’s splitting theorem for Peano arithmetic, adapted to ordered structures.

First, they introduce the notion of a bounded ultrapower. Given a dense linear order without endpoints M, a non‑principal ultrafilter U on an index set I, and the set M(I) of all functions f:I→M whose values stay within a fixed interval of M, they define an equivalence relation f∼g iff {i∈I : f(i)=g(i)}∈U. The quotient M∗=M(I)/U is equipped with the natural interpretation of symbols. For every Δ₀‑formula (i.e., a formula whose quantifiers are all bounded by intervals), Lemma 10 establishes a Łoś‑type transfer: M∗⊨φ(