论文标题
笛卡尔晶格按垂直2-SUM计数
Cartesian lattice counting by the vertical 2-sum
论文作者
论文摘要
通过删除$ l $的顶部和$ u $的底部来获得垂直2-sum的两种晶格$ l $和两个原子晶格$ u $，并通过$ u $的原子来识别$ l $的底漆。该操作根据对称情况而定，创建一个或两个非形态晶格。在这里，分析了对称情况，并提出了复发关系，该关系在某些感兴趣的家族中表达了这种垂直2-sum的数量，直到同构。分别对非同态，垂直不可分解的模块化和分布晶格进行计数和分类，分别为35和60个元素。它们的数字渐近显示为$ω（2.3122^n）$和$ω（1.7250^n）$，其中$ n $是元素的数量。显示半模块晶格的数量显示出比$ n $中任何指数的速度快。
A vertical 2-sum of a two-coatom lattice $L$ and a two-atom lattice $U$ is obtained by removing the top of $L$ and the bottom of $U$, and identifying the coatoms of $L$ with the atoms of $U$. This operation creates one or two nonisomorphic lattices depending on the symmetry case. Here the symmetry cases are analyzed, and a recurrence relation is presented that expresses the number of such vertical 2-sums in some family of interest, up to isomorphism. Nonisomorphic, vertically indecomposable modular and distributive lattices are counted and classified up to 35 and 60 elements respectively. Asymptotically their numbers are shown to be at least $Ω(2.3122^n)$ and $Ω(1.7250^n)$, where $n$ is the number of elements. The number of semimodular lattices is shown to grow faster than any exponential in $n$.
