论文标题
沃恩 - 李的nilpotent尺寸$ 12 $有限为基础
Vaughan--Lee's nilpotent loop of size $12$ is finitely based
论文作者
论文摘要
1983年，沃恩（Vaughan） - 李（Lee）表明，如果有限的nilpotent循环分为主要功率秩序因素的直接产物，那么其方程理论就有有限的基础。从那以后，是否需要直接分解的状况一直保持开放。在同一篇论文中，沃恩 - 李（Lee）给出了一个明确的示例，说明了一个订单$ 12 $的nilpotent循环，该循环不会纳入主要功率订单循环，并询问它是否有限。我们通过明确表征其术语函数来为他的示例提供有限的依据。这也使我们能够证明该循环的子功率会员资格问题可以在多项式时间内解决。
In 1983 Vaughan--Lee showed that if a finite nilpotent loop splits into a direct product of factors of prime power order, then its equational theory has a finite basis. Whether the condition on the direct decomposition is necessary has remained open since. In the same paper, Vaughan--Lee gives an explicit example of a nilpotent loop of order $12$ that does not factor into loops of prime power order and asks whether it is finitely based. We give a finite basis for his example by explicitly characterizing its term functions. This also allows us to show that the subpower membership problem for this loop can be solved in polynomial time.
