关于Burris-Willard猜想的注记
A note on the Burris-Willard conjecture
摘要：
基于Daniosečenko的结果，Burris和Willard在1987年证实，在任何$k$ -元件结构域上，其中$k\geq 3$有可能从其$k$ -ary部分双中心地产生每个中心化克隆。后来，对于每一个$k\geq 3$，斯诺构造了具有$k$ -元素载体集的代数，其中可以生成双中心化器的项操作的克隆的最小元数至少是$(k-1)^2$，其大于$k\geq 3$的$k$。我们证明了斯诺的例子不违反Burris-Willard猜想，也不使Daničenko的结果无效，后者是基于这个结果。我们还补充了我们的结果与一些计算证据$k=3$，通过一个算法来计算一个原始的正定义的关系，在一个有限集的关系克隆。
英文摘要：
Based on results by Daniľčenko, 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 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.
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