论文标题
关于部分有序集的几何概念
On a geometrical notion of dimension for partially ordered sets
论文作者
论文摘要
杜什尼克（Dushnik）和米勒（Miller）的部分顺序的众所周知的维度概念允许量化无与伦比的程度，因此被认为是部分阶订单的复杂性的量度。但是，尽管它有用，但其定义与维数的几何思想有些脱节，其中，从本质上讲，维度的数量表明需要多少个实际行来表示基础的部分有序集合。 在这里，我们介绍了dushnik-miller尺寸概念的变化，该概念更接近几何形状，Debreu维度，并显示以下主要结果：（i）如何在某些可算置限制下构建其构件，（ii）它与文献中的其他维度概念的关系，以及（III），以及在上述范围内进行的，我们可以按照上述范围进行了改进，我们可以按照上述的范围进行了改进。
The well-known notion of dimension for partial orders by Dushnik and Miller allows to quantify the degree of incomparability and, thus, is regarded as a measure of complexity for partial orders. However, despite its usefulness, its definition is somewhat disconnected from the geometrical idea of dimension, where, essentially, the number of dimensions indicates how many real lines are required to represent the underlying partially ordered set. Here, we introduce a variation of the Dushnik-Miller notion of dimension that is closer to geometry, the Debreu dimension, and show the following main results: (i) how to construct its building blocks under some countability restrictions, (ii) its relation to other notions of dimension in the literature, and (iii), as an application of the above, we improve on the classification of preordered spaces through real-valued monotones.
