Uncountable homogeneous structures

TL;DR

This paper investigates the existence and construction of uncountable homogeneous structures in model theory, introducing a new method based on the monoid of self-embeddings and Fra"iss"e theory.

Contribution

It presents a novel approach using the amalgamation property of the self-embedding monoid to construct uncountable homogeneous structures.

Findings

Established conditions for the existence of uncountable homogeneous structures.

Developed a construction method based on the monoid of self-embeddings.

Connected the existence of such structures to the amalgamation property.

Abstract

We study the existence of uncountable first-order structures that are homogeneous with respect to their finitely generated substructures. In many classical cases this is either well-known or follows from general facts, for example, if the language is finite and relational then ultrapowers provide arbitrarily large such sturctures. On the other hand, there are no general results saying that uncountable homogeneous structures with a given age exist. We examine the monoid of self-embeddings of a fixed countable homogeneous structure and, using abstract Fra\"iss\'e theory, we present a method of constructing an uncountable homogeneous structure, based on the amalgamation property of this monoid.