Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Elementary extensions of almost o-minimal structures

Mourad Berraho, Akito Tsuboi

TL;DR

The paper addresses extending an almost o-minimal structure $M$ to a proper elementary extension without losing almost o-minimality, noting that almost o-minimality is not preserved under elementary equivalence. The main construction uses bounded ultrapowers $M^{(I)}/U$ and a Gaifman-type splitting approach to obtain a proper $N \\succ M$ with $N$ almost o-minimal. It introduces generalized ultrapowers $M^{(I)^+}/U$ and $M^{(I)^-}/U$ and corresponding conditions $(\nabla)^+$ and $(\nabla)^-$ to guarantee $M \\\prec N$ while controlling definable sets on intervals. The results bridge almost o-minimality with classical tameness notions such as definable completeness and type completeness (DCTC) and provide a method to realize tame elementary extensions in the continuous setting.

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.

Elementary extensions of almost o-minimal structures

TL;DR

The paper addresses extending an almost o-minimal structure to a proper elementary extension without losing almost o-minimality, noting that almost o-minimality is not preserved under elementary equivalence. The main construction uses bounded ultrapowers and a Gaifman-type splitting approach to obtain a proper with almost o-minimal. It introduces generalized ultrapowers and and corresponding conditions and to guarantee while controlling definable sets on intervals. The results bridge almost o-minimality with classical tameness notions such as definable completeness and type completeness (DCTC) and provide a method to realize tame elementary extensions in the continuous setting.

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.
Paper Structure (5 sections, 5 theorems, 20 equations)

This paper contains 5 sections, 5 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounded ultrapowers
  4. Some generalizations
  5. Almost o-minimal structures

Key Result

Lemma 10

For all $\Delta_0$-formulas $\varphi (\bar{x})$, $\bar{x} = x_1, \dots, x_m$ and $\bar{f} = f_1, . . . , f_m \in M^{(I)}$, where $[\bar{f}] = [f_1], . . . , [f_m]$ and $\bar{f}(i) = f_1(i), \dots , f_m(i)$.

Theorems & Definitions (24)

  • Definition 1
  • Remark 2
  • Definition 5: 4
  • Example 6
  • Example 7
  • Remark 8
  • Definition 9
  • Lemma 10
  • proof
  • Theorem 11
  • ...and 14 more