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arXiv · 2011.09027

A note on the Burris-Willard conjecture

Mike Behrisch
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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.

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Mike Behrisch. 2020-11-18. A note on the Burris-Willard conjecture. https://arxiv.org/abs/2011.09027

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