Extensive embeddings into Fra\"iss\'e structures and stationary weak independence relations

TL;DR

This paper explores the relationship between stationary weak independence relations and extensible embeddings in Fra"issé structures, showing that certain ordered structures with SWIR have extensible $ ext{ω}$-age, and examining various examples.

Contribution

It establishes a connection between SWIR and extensible $ ext{ω}$-age in Fra"issé structures and provides examples illustrating when these properties hold or do not.

Findings

Linearly ordered Fra"issé structures with SWIR have extensible ω-age.

Examples show that the two properties can occur independently.

Most countably infinite ultrahomogeneous oriented graphs have extensible ω-age.

Abstract

Let $M$ be a Fra\"iss\'e structure (a countably infinite ultrahomogeneous structure). We call an embedding $f : A \to M$ extensive if each automorphism of its image extends to an automorphism of $M$, where the extension map respects composition, and we say that $M$ has extensible $\omega$-age if each substructure admits an extensive embedding into $M$. We investigate the relationship between the following two properties: the presence of a stationary weak independence relation (SWIR) on $M$, and extensibility of the $\omega$-age of $M$. We show that linearly ordered Fra\"iss\'e structures with a SWIR have extensible $\omega$-age, but also we give examples of Fra\"iss\'e structures where only one of the two properties holds. Finally, we consider whether a wide range of examples of Fra\"iss\'e structures have extensible $\omega$-age or a finite SWIR expansion, including all countably infinite ultrahomogeneous oriented graphs (with one exception).