A note on the Burris-Willard conjecture

Abstract

Based on results by Daniľčenko, in 1987 Burris and Willard have conjectured that on any k-element domain where k≥3 it is possible to bicentrically generate every centraliser clone from its k-ary part. Later, for every k≥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≥3. We prove that Snow's examples do not violate the Burris-Willard conjecture nor invalidate the results by Daniľčenko 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.