Congruence normality: The characterization of the doubling class of convex sets
Congruence normality: The characterization of the doubling class of convex sets
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同余正态性：凸集加倍类的表征
DOI：
10.1007/bf01221793
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发表时间：
1994
影响因子：
0.6
通讯作者：
A. Day
中科院分区：
文献类型：
--
作者：
A. Day
在[2]中，作者给出了所有下有界、上有界或全有界（有限）格的类在倍增构造上的刻画：即它们都是有限格，从1开始，分别用下伪区间、上伪区间或（普通）区间在倍增构造下闭合类。最近，在[5]中，Geyer给出了所有凸集的“加倍类”的一个表征。然而，Geyer的表征和证明是用概念格的语言写的，因此传统的格概念的作用，例如连接-相遇-不可约，同余关系等，是高度不可见的。在本文中，我们给出了该类的另一个特征，我们称之为同余正规，并提供了盖耶结果的传统证明。我们还将尝试通过提供该类的一些“好”性质来证明“同余法线”一词的使用是正确的。特别地，我们对Geyer的结果进行了改进，以获得更低的响应。上，有界（有限）格作为同余法格满足连接-，对应。满足-，半分配律（SDv），参见（SD^）。
In [2], the author gave characterizations of the classes of all lower, upper or fully bounded (finite) lattices in terms of the doubling construction: viz, they are all finite lattices attainable by starting with 1 and closing the class under the doubling construction using lower pseudo-, upper pseudo-or (ordinary) intervals, respectively. Recently, in [5], Geyer provided a characterization of the" doubling class" of all convex sets. Geyer's characterization and proof however are written in the language of concept lattices, and therefore the roles of traditional lattice concepts, eg join-and meet-irreducibles, congruence relations, etc., are highly non-visible. In this paper we present another characterization of this class, which we call congruence normal, and provide traditional proofs of Geyer's results. We will also attempt to justify the use of the term, congruence normal, by supplying some" good" properties of this class. In particular, we refine Geyer's results to obtain lower, resp. upper, bounded (finite) lattices as congruence normal lattices satisfying the join-, resp. meet-, semidistributive law (SDv), resp.(SD^).