The "minimum" of a class with respect to
is its maximum with respect to the converse of
; and the "lower limit" with respect to
is the upper limit with respect to the converse of
.
The notions of limit and maximum do not essentially demand that the relation in respect to which they are defined should be serial, but they have few important applications except to cases when the relation is serial or quasi-serial. A notion which is often important is the notion "upper limit or maximum," to which we may give the name "upper boundary." Thus the "upper boundary" of a set of terms chosen out of a series is their last member if they have one, but, if not, it is the first term after all of them, if there is such a term. If there is neither a maximum nor a limit, there is no upper boundary. The "lower boundary" is the lower limit or minimum.