set theory - What are interesting families of subsets of a given set?

Another ultrafilter cousin is the concept of a majority
space. This is a family $M$ of nonempty subsets of $X$,
called the majorities, such that any superset of a
majority is a majority, every subset of $X$ or its
complement is a majority, and if disjoint sets are
majorities, then they are complements. A strict majority
space has $Yin Mto Y^cnotin M$, and otherwise they are
called weak majorities. A vast majority space is closed
under finite differences in majorities. There are other
various overwhelming majority concepts.

The main point is that the majority space concept
generalizes the ultrafilter concept by omitting the
intersection rule. Every ultrafilter on $X$ is a majority
space. But there are others. For example, on a finite set,
one may take the the subsets with at least half the size, and of course this situation motivates the voting theory terminology.

On an infinite set $X$, one can divide it into a finite odd
number of disjoint pieces $X_i$, each carrying an
ultrafilter $mu_i$ on $X_i$, and then saying that
$Ysubset X$ is a majority if for most $i$ one has
$Ycap X_iinmu_i$. This produces a vast majority on
$X$ that is not an ultrafilter.

Eric Pacuit has investigated majority logic, and I recall that Andreas Blass has some very interesting work showing that it is consistent with ZFC that every majority space derives from ultrafilters in a simple way.