The lattice of clones of self-dual operations collapsed

TL;DR

This paper characterizes the structure of clones of self-dual operations on a three-element set, revealing a stark contrast between the uncountably many clones under clone homomorphisms and only countably many under minor-preserving maps.

Contribution

It provides a complete classification of self-dual operation clones ordered by minor-preserving maps, connecting clone theory with primitive positive constructability.

Findings

Continuum many clones under clone homomorphisms.

Countably many clones under minor-preserving maps.

Full description of the lattice of self-dual operation clones.

Abstract

There are continuum many clones on a three-element set even if they are considered up to \emph{homomorphic equivalence}. The clones we use to prove this fact are clones consisting of \emph{self-dual operations}, i.e., operations that preserve the relation $\{(0,1),(1,2),(2,0)\}$. However, there are only countably many such clones when considered up to equivalence with respect to \emph{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, ordered by primitive positive constructability, because there is a minor-preserving map from the polymorphism clone of a finite structure $\mathfrak A$ to the polymorphism clone of a finite structure $\mathfrak B$ if and only if there is a primitive positive construction of $\mathfrak B$ in $\mathfrak A$.