On polymorphism-homogeneous relational structures and their clones

TL;DR

This paper explores the concept of polymorphism-homogeneity in relational structures, linking it to existing algebraic notions, and provides characterizations and decidability results for various classes of structures.

Contribution

It introduces and studies polymorphism-homogeneity, connecting it to algebraic and model-theoretic properties, and offers classifications and decidability results for specific structures.

Findings

Polymorphism-homogeneity is equivalent to quantifier elimination for positive primitive formulae in weakly oligomorphic structures.

Decidability of polymorphism-homogeneity for finite structures using Baker-Pixley theorem.

Complete classifications of countable polymorphism-homogeneous graphs and posets.

Abstract

A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. This notion was introduced recently by Cameron and Ne\v{s}et\v{r}il. In this paper we consider a strengthening of homomorphism-homogeneity --- we call a relational structure polymorphism-homogeneous if every partial polymorphism with a finite domain extends to a global polymorphism of the structure. It turns out that this notion (under various names and in completely different contexts) has been existing in algebraic literature for at least 30 years. Motivated by this observation, we dedicate this paper to the topic of polymorphism-homogeneous structures. We study polymorphism-homogeneity from a model-theoretic, an algebraic, and a combinatorial point of view. E.g., we study structures that have quantifier elimination for positive primitive formulae, and show that this notion is equivalent to polymorphism-homogeneity for weakly oligomorphic structures. We demonstrate how the Baker-Pixley theorem can be used to show that polymorphism-homogeneity is a decidable property for finite relational structures. Eventually, we completely characterize the countable polymorphism-homogeneous graphs, the polymorphism-homogeneous posets of arbitrary size, and the countable polymorphism-homogeneous strict posets.