Homogenizable structures and model completeness

TL;DR

This paper explores the concept of homogenizable structures, classifies them into different categories, and establishes a key link between model completeness and homogenizability, providing a precise characterization for omega-categorical structures.

Contribution

It introduces a classification of homogenizable structures, highlights the importance of model completeness, and offers a necessary and sufficient condition for omega-categorical structures to be homogenizable.

Findings

Model completeness is crucial for the relation between structures and amalgamation bases.

Provides a necessary and sufficient condition for omega-categorical model-complete structures to be homogenizable.

Classifies various known examples of homogenizable structures into different categories.

Abstract

A homogenizable structure $\mathcal{M}$ is a structure where we may add a finite amount of new relational symbols to represent some $\emptyset-$definable relations in order to make the structure homogeneous. In this article we will divide the homogenizable structures into different classes which categorize many known examples and show what makes each class important. We will show that model completeness is vital for the relation between a structure and the amalgamation bases of its age and give a necessary and sufficient condition for an $\omega-$categorical model-complete structure to be homogenizable.