The Amalgamation Property for automorphisms of ordered abelian groups

TL;DR

The paper proves that the category of ordered abelian groups with an automorphism has the Amalgamation Property, enabling a tame positive-logic analysis of existentially closed structures. It introduces a realification framework that lifts oag automorphisms to automorphisms of ordered real vector spaces, and establishes an Intermediate Value Property for σ-polynomials in existentially closed models. By combining IVP with AP, the authors deduce that the positive theory of automorphisms of oags is NIP, and they develop a positive-logic toolkit (Pillay/Robinson theories, coheirs, invariant types) to study tameness beyond first-order logic. The results yield a generalised Hahn embedding perspective and provide a foundation for further development of NIP in positive theories, with potential implications for valued fields and automorphism-influenced algebraic structures.

Abstract

We prove that the category of ordered abelian groups equipped with an automorphism has the Amalgamation Property, deduce that their inductive theory is NIP in the sense of positive logic, and initiate a development of the latter framework. As byproducts of the proof, we obtain a generalised version of the Hahn Embedding Theorem which allows to lift each automorphism of an ordered abelian group to one of an ordered real vector space, and we show that, on existentially closed structures, linear combinations of iterates of the automorphism have the Intermediate Value Property.

Figures (3)

• Figure 1: An amalgamation diagram of automorphisms of oags.
• Figure 2: The diagram constructed in the proof of \ref{['lemma:algap']}.
• Figure 3: The diagram constructed in the proof of \ref{['thm:apqtor']}.

Theorems & Definitions (167)

• Theorem A: \ref{['thm:apqtor']}
• Theorem B: \ref{['thm:redtoR']}
• Theorem C: \ref{['thm:minima']}
• Theorem D: \ref{['co:apthennip']}
• Example 1.1
• Definition 1.3
• Remark 1.5
• Definition 1.6
• Example 1.7
• Example 1.9