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Concept
Ultrafilter
Summary
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") is a certain subset of namely a maximal filter on that is, a proper filter on that cannot be enlarged to a bigger proper filter on
If is an arbitrary set, its power set ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on are usually called . An ultrafilter on a set may be considered as a finitely additive measure on . In this view, every subset of is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0), depending on whether it belongs to the given ultrafilter or not.
Ultrafilters have many applications in set theory, model theory, topology and combinatorics.
In order theory, an ultrafilter is a subset of a partially ordered set that is maximal among all proper filters. This implies that any filter that properly contains an ultrafilter has to be equal to the whole poset.
Formally, if is a set, partially ordered by then
a subset is called a filter on if
is nonempty,
for every there exists some element such that and and
for every and implies that is in too;
a proper subset of is called an ultrafilter on if
is a filter on and
there is no proper filter on that properly extends (that is, such that is a proper subset of ).
Every ultrafilter falls into exactly one of two categories: principal or free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form for some (but not all) elements of the given poset. In this case is called the of the ultrafilter. Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter.
For ultrafilters on a powerset a principal ultrafilter consists of all subsets of that contain a given element Each ultrafilter on that is also a principal filter is of this form. Therefore, an ultrafilter on is principal if and only if it contains a finite set.
Official source
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