A note on the Burris-Willard conjecture

TL;DR

This paper examines the Burris-Willard conjecture on generating centraliser clones from their k-ary parts, clarifies that Snow's counterexamples do not disprove it, and provides computational evidence for the case k=3.

Contribution

The paper proves Snow's examples do not violate the Burris-Willard conjecture and offers computational evidence for the case k=3.

Findings

Snow's examples do not violate the conjecture

The conjecture holds for the examined cases

Computational methods support the conjecture for k=3

Abstract

Based on results by Dani\v{l}\v{c}enko, in 1987 Burris and Willard have conjectured that on any $k$-element domain where $k\geq 3$ it is possible to bicentrically generate every centraliser clone from its $k$-ary part. Later, for every $k\geq 3$, Snow constructed algebras with a $k$-element carrier set where the minimum arity of the clone of term operations from which the bicentraliser can be generated is at least $(k-1)^2$, which is larger than $k$ for $k\geq 3$. We prove that Snow's examples do not violate the Burris-Willard conjecture nor invalidate the results by Dani\v{l}\v{c}enko on which the latter is based. We also complement our results with some computational evidence for $k=3$, obtained by an algorithm to compute a primitive positive definition for a relation in a finitely generated relational clone over a finite set.