Projective clone homomorphisms

TL;DR

This paper explores the connection between clone homomorphisms to projection clones and their continuity, particularly in the context of countable ω-categorical structures interpreting all finite structures primitively positively.

Contribution

It investigates when clone homomorphisms to projection clones are continuous, clarifying conditions for interpreting finite structures primitively positively.

Findings

Continuous clone homomorphisms imply interpretability of all finite structures.

The existence of a clone homomorphism to the projection clone is characterized.

Continuity of clone homomorphisms is crucial for certain interpretability properties.

Abstract

It is known that a countable $\omega$-categorical structure interprets all finite structures primitively positively if and only if its polymorphism clone maps to the clone of projections on a two-element set via a continuous clone homomorphism. We investigate the relationship between the existence of a clone homomorphism to the projection clone, and the existence of such a homomorphism which is continuous and thus meets the above criterion.