论文标题
四元素生成的分区晶格及其直接产品
Four-element generating sets of partition lattices and their direct products
论文作者
论文摘要
令$ n> 3 $为自然数字。到1975年H. Strietz的结果，$ n $ element套件的所有分区的晶格部分$（n）$（n）具有四元素的生成集。 1983年，L。Zádori以特别优雅的结构给出了这一事实的新证明。根据他1983年的建设，本文对四元素生成的$ν（n）$ part $（n）$的数量$ν（n）$进行了下限。我们还为$ n $的$ν（n）$提供了计算机辅助统计方法。 L.Zádori在1983年的论文中还证明，对于$ n \ geq 7 $，晶格部分$（n）$具有四个元素生成集，而不是抗抗小节。他留下了问题，无论是否存在为$ n \ in \ {5,6 \} $的$ n \的生成集。在这里，我们以$ n = 5 $的否定解决此问题，并以$ n = 6 $的肯定解决。 最后，主要定理断言，某些分区晶格的直接产物是四生的。特别是，到本定理的第一部分，部分$（n_1）\ times $ part $（n_2）$对于任何两个不同的整数$ n_1 $和$ n_2 $至少为5。定理的第二部分是技术的。也就是说，直接产品部分$（n）$ $ \ times $ part $（n+1）$ $ \ times \ times \ times \ times $ part $ $（3n-14）$是针对每个整数$ n \ geq 9 $的四生。另外，对于每一个正整数$ u $，$ u $ - 直接产品部分的直接功率$（n）$ $ \ times $ part $ $（n+1）$ $ \ times \ times \ times \ times $ part $ part $（n+u-1）$是为所有$ n $ n $ n $ of的四个生成的。如果我们不坚持过多的直接因素，那么指数可能很大。例如，我们的定理意味着$ 10^{127} $ - 部分$ $（1011）$ $ $ \ times $ part $（1012）$ $ $ \ times \ times \ times \ times \ times $ part $（2020）$是四生。
Let $n>3$ be a natural number. By a 1975 result of H. Strietz, the lattice Part$(n)$ of all partitions of an $n$-element set has a four-element generating set. In 1983, L. Zádori gave a new proof of this fact with a particularly elegant construction. Based on his construction from 1983, the present paper gives a lower bound on the number $ν(n)$ of four-element generating sets of Part$(n)$. We also present a computer assisted statistical approach to $ν(n)$ for small values of $n$. In his 1983 paper, L. Zádori also proved that for $n\geq 7$, the lattice Part$(n)$ has a four element generating set that is not an antichain. He left the problem whether such a generating set for $n\in\{5,6\}$ exists open. Here we solve this problem in negative for $n=5$ and in affirmative for $n=6$. Finally, the main theorem asserts that the direct product of some powers of partition lattices is four-generated. In particular, by the first part of this theorem, Part$(n_1)\times$ Part$(n_2)$ is four-generated for any two distinct integers $n_1$ and $n_2$ that are at least 5. The second part of the theorem is technical but it has two corollaries that are easy to understand. Namely, the direct product Part$(n)$ $\times$ Part$(n+1)$ $\times\dots\times$ Part$(3n-14)$ is four-generated for each integer $n\geq 9$. Also, for every positive integer $u$, the $u$-th the direct power of the direct product Part$(n)$ $\times$ Part$(n+1)$ $\times\dots\times$ Part$(n+u-1)$ is four-generated for all but finitely many $n$. If we do not insist on too many direct factors, then the exponent can be quite large. For example, our theorem implies that the $10^{127}$-th direct power of Part$(1011)$ $\times$ Part$(1012)$ $\times \dots \times$ Part$(2020)$ is four-generated.
