The amalgamation property and Urysohn structures in continuous logic

TL;DR

This paper explores the conditions under which classes of continuous structures have the amalgamation property and the existence of Urysohn structures, linking these to automorphism groups and universal groups in continuous logic.

Contribution

It characterizes the moduli of continuity for amalgamation, establishes the existence of Urysohn structures, and proves Fra"issé properties and universal automorphism groups in continuous logic.

Findings

Characterized moduli of continuity for amalgamation.

Proved existence of Urysohn continuous structures.

Established Fra"issé classes and universal automorphism groups.

Abstract

In this paper we consider the classes of all continuous $\mathcal{L}$-(pre-)structures for a continuous first-order signature $\mathcal{L}$. We characterize the moduli of continuity for which the classes of finite, countable, or all continuous $\mathcal{L}$-(pre-)structures have the amalgamation property. We also characterize when Urysohn continuous $\mathcal{L}$-(pre)-structures exist, establish that certain classes of finite continuous $\mathcal{L}$-structures are countable Fra\"iss\'e classes, prove the coherent EPPA for these classes of finite continuous $\mathcal{L}$-structures, and show that actions by automorphisms on finite $\mathcal{L}$-structures also form a Fra\"iss\'e class. As consequences, we have that the automorphism group of the Urysohn continuous $\mathcal{L}$-structure is a universal Polish group and that Hall's universal locally finite group is contained in the automorphism group of the Urysohn continuous $\mathcal{L}$-structure as a dense subgroup.