论文标题
$ \ Mathcal C_1 $ -DIAGRAM SLIM矩形半模块晶格允许商图
$\mathcal C_1$-diagrams of slim rectangular semimodular lattices permit quotient diagrams
论文作者
论文摘要
G.Grätzer和E. Knapp在2007年和2009年介绍了细滑的半中型格（简短，SPS晶格）和细长的矩形晶格（简而言之，Sr晶格）。这些晶格必然是有限的和平面的。自2007年以来，它们是在2007年以来的$ nose $。作者于2017年提出。对于有限晶格$ l $的图$ f $和$ l $的一致性$α$，我们通过占据''$ f/α$来定义'$ f/α$，通过占据$α$ blocks的最大元素并保留其几何位置。虽然$ f/α$通常都不是Hasse图，但我们证明，每当$ l $是Sr lattice，而$ f $是$ \ MATHCAL C_1 $ -DIAGRAM的$ l $，那么$ f/α$就是$ \ Mathcal C_1 $ diagram of $ l/α$的diagram of $ l/α$，这是Sr lattice或sr lattice sh lattice。在服用过滤器下，晶格的同构成晶格的晶格同构型晶格已关闭。我们证明，此类在另外两个结构下是关闭的，这些结构是在某种意义上进行过滤器的媒介。相应的两个证据之一依赖于商图构造的倒数。
Slim semimodular lattices (for short, SPS lattices) and slim rectangular lattices (for short, SR lattices) were introduced by G. Grätzer and E. Knapp in 2007 and 2009. These lattices are necessarily finite and planar, and they have been studied in more then four dozen papers since 2007. They are best understood with the help of their $\mathcal C_1$-diagrams, introduced by the author in 2017. For a diagram $F$ of a finite lattice $L$ and a congruence $α$ of $L$, we define the ``quotient diagram'' $F/α$ by taking the maximal elements of the $α$-blocks and preserving their geometric positions. While $F/α$ is not even a Hasse diagram in general, we prove that whenever $L$ is an SR lattice and $F$ is a $\mathcal C_1$-diagram of $L$, then $F/α$ is a $\mathcal C_1$-diagram of $L/α$, which is an SR lattice or a chain. The class of lattices isomorphic to the congruence lattices of SPS lattices is closed under taking filters. We prove that this class is closed under two more constructions, which are inverses of taking filters in some sense; one of the two respective proofs relies on an inverse of the quotient diagram construction.
