超限有向图
Transfinite Digraphs
摘要：
超限图已经被定义和检查在各种先前的工作，但超限有向图尚未被研究。目前的工作就是从这样一个任务出发的。与序数一样，超限有向图出现在由可数序数索引的等级层次中。秩为0的有向图是常规有向图。秩为1的图是通过定义0-秩有向图的某些端点，然后划分这些端点以获得秩为1的顶点来构造的。然后，秩为0的有向图在秩为1的那些顶点处连接在一起以获得秩为1的有向图。这个过程可以通过自然数等级继续进行。然而，为了实现秩是第一无限序数$ω$的有向图（即，第一极限序数），这是一种特殊的超限有向图，我们称之为具有“箭秩”的有向图，它必须首先以不同于自然数秩的方式构造.然后，更高秩的有向图可以以类似于自然数秩有向图的方式构造。然而，就在每个极限序秩之前，必须建立一个箭头秩的有向图。
英文摘要：
Transfinite graphs have been defined and examined in a variety of prior works, but transfinite digraphs had not as yet been investigated. The present work embarks upon such a task. As with the ordinals, transfinite digraphs appear in a hierarchy of ranks indexed by the countable ordinals. The digraphs of rank 0 are the conventional digraphs. Those of rank 1 are constructed by defining certain extremities of 0-ranked digraphs, and then partitioning those extremities to obtain vertices of rank 1. Then, digraphs of rank 0 are connected together at those vertices of rank 1 to obtain a digraph of rank 1. This process can be continued through the natural-number ranks. However, to achieve a digraph whose rank is the first infinite ordinal $ω$ (i.e., the first limit ordinal), a special kind of transfinite digraph, which we call a digraph with an "arrow rank" must first be constructed in a way different from those of natural-number rank. Then, digraphs of still higher ranks can be constructed in a way similar to that for the natural-number ranked digraphs. However, just before each limit-ordinal rank, a digraph of arrow rank must be set up.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
