Is it true in general that a filter is given by the intersection of the ultrafilters refining it?

Question

In set theory it holds that any filter is the intersection of all the ultrafilters refining it. By the way the definition of filter can be given in a more general context, that is as a particular subset of a partially ordered set. The set-case is then a particular example, taking as poset the powerset and as order relation the inclusion. See http://en.wikipedia.org/wiki/Filter_(mathematics). I was wondering if the above property is still valid in such a general situation, or at least for the case of a principal filter. Thank you.

Answer 1

For general posets we have to review the usual definition of ultrafilter (a filter where for every set either the set or its complement is in the filter; outside the set-setup we don't have the notion of complement), e.g. say that it is a maximal proper filter (i.e. any larger filter is the whole poset). However, with the poset $(\{1,2,3,4\},\le)$, we have only one maximal proper filter $U=\{2,3,4\}$, hence the filter $F=\{3,4\}$ is not the intersection of its containing ultrafilters.