论文标题
关于1960年部分补体的算法结构
On the algorithmic construction of the 1960 sectional complement
论文作者
论文摘要
1960年，G。Grätzer和E。\，t。施密特证明，每个有限的分布晶格都可以表示为截面有限晶格$ l $的一致性晶格。对于$ u \ leq v $在$ l $中，他们构建了一个部分补充，现在称为\ emph {1960截面补充}。 1999年，G.Grätzer和e。\，t。施密特（Schmidt）发现了一种非常简单的方法，可以在一个切碎的晶格的理想晶格中构建截面补充，由两个截面互补的有限晶格组成，仅在两个元素（原子引理）中重叠。提出了这个问题，是否可以将这个简单的过程推广到找到1960年部分补体的算法。 2006年，G。〜Grätzer和M. Roddy发现了这样的算法 - 允许其进行宽大的纬度。 在本文中，我们证明算法中明显的宽大纬度具有欺骗性：无论采用哪种方式执行算法，它都会产生相同的截面补体。实际上，这解决了Grätzer-Roddy纸的问题2和3。令人惊讶的是，该算法提供的独特部分补体是1960年的部分补充，解决了同一论文的问题1。
In 1960, G. Grätzer and E.\,T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of a sectionally complemented finite lattice $L$. For $u \leq v$ in $L$, they constructed a sectional complement, which is now called the \emph{1960 sectional complement}. In 1999, G. Grätzer and E.\,T. Schmidt discovered a very simple way of constructing a sectional complement in the ideal lattice of a chopped lattice made up of two sectionally complemented finite lattices overlapping in only two elements -- the Atom Lemma. The question was raised whether this simple process can be generalized to an algorithm that finds the 1960 sectional complement. In 2006, G.~Grätzer and M. Roddy discovered such an algorithm -- allowing a wide latitude how it is carried out. In this paper we prove that the wide latitude apparent in the algorithm is deceptive: whichever way the algorithm is carried out, it~produces the same sectional complement. This solves, in fact, Problems 2 and 3 of the Grätzer-Roddy paper. Surprisingly, the unique sectional complement provided by the algorithm is the 1960 sectional complement, solving Problem 1 of the same paper.
