论文标题
关于戴维斯子集晶格，原子晶格和T1分离公理下的闭合系统之间的隐态性
On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom
论文作者
论文摘要
在本文中，我们将所有单个元素集都关闭时，计算了设置闭合系统（也称为Moore家族）。特别是，我们为尺寸$ n = 6 $的基本组提供了如此严格的（包括空集）和非图案的家庭数量。我们还提供了此类不等的摩尔家庭的数量，就设置为$ n = 6 $的所有基础排列的数量。在OEI和现有文献中的搜索揭示了发现的数字与D. \ M.〜Davis设置的工会晶格（\ seqnum {a235604}的条目，最高可达$ n = 5 $）和$ | \ Mathcal l_n | $ n | $ n $ n $ n $ n $ up up n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $。因此，我们研究了所有这些情况，通过GALOIS辅助和正式概念分析建立它们之间的一对一对应关系，并为读者提供我们的两种枚举算法以及用于其他测试的这些算法的结果。其他结果包括$ n = 6 $的最大交叉点家庭尺寸，加上我们的猜想$ n = 7 $，是原子晶格$ \ Mathcal l_n $的上限，以及基于极端晶格理论的$ \ Mathcal l_n $的某些结构属性。
In this paper we count set closure systems (also known as Moore families) for the case when all single element sets are closed. In particular, we give the numbers of such strict (empty set included) and non-strict families for the base set of size $n=6$. We also provide the number of such inequivalent Moore families with respect to all permutations of the base set up to $n=6$. The search in OEIS and existing literature revealed the coincidence of the found numbers with the entry for D.\ M.~Davis' set union lattice (\seqnum{A235604}, up to $n=5$) and $|\mathcal L_n|$, the number of atomic lattices on $n$ atoms, obtained by S.\ Mapes (up to $n=6$), respectively. Thus we study all those cases, establish one-to-one correspondences between them via Galois adjunctions and Formal Concept Analysis, and provide the reader with two of our enumerative algorithms as well as with the results of these algorithms used for additional tests. Other results include the largest size of intersection free families for $n=6$ plus our conjecture for $n=7$, an upper bound for the number of atomic lattices $\mathcal L_n$, and some structural properties of $\mathcal L_n$ based on the theory of extremal lattices.
