Last Updated：2022/12/24
An ultrafilter is maximal in the sense that if any other element of the poset not already in it were added to it, one could deduce (from the laws which define the filter, and the given ordering relation, i.e., the structure of the poset) that the resulting filter must be improper; i.e., it must contain all the elements of the poset.
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An
ultrafilter
is
maximal
in
the
sense
that
if
any
other
element
of
the
poset
not
already
in
it
were
added
to
it,
one
could
deduce
(from
the
laws
which
define
the
filter,
and
the
given
ordering
relation,
i.e.,
the
structure
of
the
poset)
that
the
resulting
filter
must
be
improper;
i.e.,
it
must
contain
all
the
elements
of
the
poset.
