The Completion of a Partially Ordered Set with Respect to a Polarization
The Completion of a Partially Ordered Set with Respect to a Polarization
复制标题
相对于极化的偏序集的补全
DOI：
10.1112/plms/s3-28.1.13
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发表时间：
1974
影响因子：
1.8
通讯作者：
W. R. Tunnicliffe
中科院分区：
文献类型：
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作者：
W. R. Tunnicliffe
1.引言一个偏序集(偏序集)X的一个完全扩张是一个对(C，j)，其中C是一个完备格，j：X->；C是一个序嵌入。补全是一个不太大的完整扩展，在§2中被形式化。从MacNeille([10])的构造可知，每个偏序集至少有一个补全。例如，在[1]、[5]、[9]和[16]中，第47页已经给出了构造完全扩张的其他方法，并且在[6]中给出了包括所有这些的更一般的方法(见示例2)。然而，这些构造都不会生成非常广泛的扩展类。例如，它们不包括像在封闭单位正方形中嵌入开放单位正方形那样简单而重要的结构(如果x<；x‘和y<；y’，则每个按(x，y)^(%‘，y’)排序)。这里描述了一种构造，它生成比任何早期方法都要宽得多的扩展类(尽管给出的例子表明，不是所有偏序集的完备化都可以通过这种方式获得)。该方法基于极化(定义3)的思想，极化是一种可以在任何点集X中引入的结构。极化是X的子集的有序对(满足两个简单条件)。与X的每个极化0&>相联系，构造了一个完备格M(X，0->t；)和X在M(X，&)中的一个注入I9(§3)。在定理1(§4)中，给出了偏序集X的极化0*使得(M(X，SP)，I&)是X的完备化的一个充要条件。这样的极化被称为一致(定义5)，并且对{M{X，&)，IP)表示完成
1. Introduction A complete extension of a poset (partially ordered set) X is a pair (C, j), where C is a complete lattice and j : X-> C is an order-embedding. A completion is a complete extension that is 'not too large', in a sense formalized in § 2. That every poset has at least one completion is known from a construction of MacNeille ([10]). Other methods of constructing complete extensions have been given for, example, in [1], [5], [9], and [16], p. 47, and a more general method including all these in [6] (see Example 2). However, none of these constructions generates a very wide class of extensions. They do not, for example, include so simple and important a construction as the embedding of the open unit square in the closed unit square (each being ordered by (x, y) ^ (%',y') if x < x' and y < y'). Here a construction is described that generates a considerably wider class of extensions than does any earlier method (although an example is given that shows that not every completion of a poset can be obtained this way). The method is based on the idea of a polarization (Definition 3), which is a structure that can be introduced in any point set, X. A polarization is an ordered pair of sets of subsets of X (satisfying two simple conditions). Associated with each polarization 0> of X, a complete lattice M(X, 0>) and an injection i 9 of X in M(X,&) are constructed (§3). In Theorem 1 (§4), a necessary and sufficient condition on a polarization 0* of a poset X so that (M(X, SP), i &) is a completion of X is given. Such a polarization is called consistent (Definition 5), and the pair {M{X,&),ip) the completion