We now lay down the following definition:—
A relation is serial when it is an aliorelative, transitive, and connected; or, what is equivalent, when it is asymmetrical, transitive, and connected.
A series is the same thing as a serial relation.
It might have been thought that a series should be the field of a serial relation, not the serial relation itself. But this would be an error. For example,
are six different series which all have the same field. If the field were the series, there could only be one series with a given field. What distinguishes the above six series is simply the different ordering relations in the six cases. Given the ordering relation, the field and the order are both determinate. Thus the ordering relation may be taken to be the series, but the field cannot be so taken.
Given any serial relation, say
, we shall say that, in respect of this relation,
