Relativised homomorphism preservation at the finite level

TL;DR

This paper explores the limits of the homomorphism preservation property in finite structures, showing failures in lattices but establishing a relativised preservation theorem for relational structures, connecting to constraint satisfaction problems.

Contribution

It provides a complete finite-level characterisation of first-order definable anti-varieties relative to relational classes, extending homomorphism preservation results.

Findings

Homomorphism-closed classes of finite lattices are not always definable by existential positive sentences.

A finite-level relativised homomorphism preservation theorem is established for relational structures.

Connections to Atserias's characterisation of first-order definable constraint satisfaction problems.

Abstract

In this article, we investigate the status of the homomorphism preservation property amongst restricted classes of finite relational structures and algebraic structures. We show that there are many homomorphism-closed classes of finite lattices that are definable by a first-order sentence but not by existential positive sentences, demonstrating the failure of the homomorphism preservation property for lattices at the finite level. In contrast to the negative results for algebras, we establish a finite-level relativised homomorphism preservation theorem in the relational case. More specifically, we give a complete finite-level characterisation of first-order definable finitely generated anti-varieties relative to classes of relational structures definable by sentences of some general forms. When relativisation is dropped, this gives a fresh proof of Atserias's characterisation of first-order definable constraint satisfaction problems over a fixed template, a well known special case of Rossman's Finite Homomorphism Preservation Theorem.