Cores of Countably Categorical Structures

TL;DR

This paper proves that every countably categorical structure has a unique core, which is either finite or countably categorical, impacting the classification of constraint satisfaction problems.

Contribution

It establishes the existence and uniqueness of cores for all -structures and their homomorphic equivalence to model-complete cores, extending core theory beyond finite structures.

Findings

Every -structure has a core.

Existence of a homomorphically equivalent model-complete core.

Implications for -structure-based constraint satisfaction problems.

Abstract

A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure has a core, i.e., has an endomorphism such that the structure induced by its image is a core; moreover, the core is unique up to isomorphism. Weprove that every \omega -categorical structure has a core. Moreover, every \omega-categorical structure is homomorphically equivalent to a model-complete core, which is unique up to isomorphism, and which is finite or \omega -categorical. We discuss consequences for constraint satisfaction with \omega -categorical templates.