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 $k\ge 3$ it is possible to bicentrically generate every centraliser clone from its $k$-ary part. Later, for every $k\ge 3$, 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 $k\ge 3$. 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.

24 pages, contains an ancillary directory containing implementations in c++ and z3-scripts verifying parts of the paper