The lattice of clones of self-dual operations collapsed

Abstract

We prove that there are continuum many clones on a three-element set even if they are considered up to homomorphic equivalence. The clones we use to prove this fact are clones consisting of self-dual operations, i.e. operations that preserve the relation [Formula: see text]. However, there are only countably many such clones when considered up to equivalence with respect to minor-preserving maps instead of clone homomorphisms. We give a full description of the set of clones of self-dual operations, ordered by the existence of minor-preserving maps. Our result can also be phrased as a statement about structures on a three-element set: we give a full description of the structures containing the relation [Formula: see text], ordered by primitive positive constructability, because there is a minor-preserving map from the polymorphism clone of a finite structure [Formula: see text] to the polymorphism clone of a finite structure [Formula: see text] if and only if there is a primitive positive construction of [Formula: see text] in [Formula: see text].