Uncountable homogeneous structures
arXiv:2411.17889 · doi:10.1016/j.apal.2025.103649
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 Fraïssé theory, we present a method of constructing an uncountable homogeneous structure, based on the amalgamation property of this monoid.
An earlier version of this paper had three authors. Mirna Džamonja withdrew from the collaboration on the article on January 24, 2025. See pg. 23 for the acknowledgment