and
cannot be the same person; but it is not true that of any two persons one must be an ancestor of the other.
The question of the circumstances under which series can be generated by ancestral relations derived from relations of consecutiveness is often important. Some of the most important cases are the following: Let
be a many-one relation, and let us confine our attention to the posterity of some term
. When so confined, the relation "proper
-ancestor" must be connected; therefore all that remains to ensure its being serial is that it shall be contained in diversity. This is a generalisation of the instance of the dinner-table. Another generalisation consists in taking