  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 $\omega$ (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.
