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Another way of looking at ultrafilters on a set S is to define a function m on the power set of S by setting m(A) = 1 if A is contained in F and m(A) = 0 otherwise. Then m is a finitely additive measure on S, and every property of elements of S is either true almost everywhere or false almost everywhere.
There are two very different types of ultrafilter: principal and free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form Fa={x | a≤x} for some (but not all) elements a of the given poset. In this case a is called the principal element of the ultrafilter. For the case of filters on sets, the elements that qualify as principals are exactly the one-element sets. Thus, a principal ultrafilter on a set S consists of all sets containing a particular point of S. Any ultrafilter which is not principal is called a free (or non-principal) ultrafilter.
One can show that every filter is contained in an ultrafilter (see Ultrafilter Lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice in the form of Zorn's Lemma. Consequently explicit examples of free ultrafilters cannot be given. Nonetheless, almost all ultrafilters on an infinite set are free. By contrast, every ultrafilter of a finite poset (or on a finite set) is principal, since any finite filter has a least element.
Ultrafilters on sets are useful in topologyTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g, especially in relation to compactIn mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R n in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if e HausdorffIn topology and related branches of mathematics, Hausdorff spaces and preregular spaces are kinds of topological spaces. The conditions imposed are the most significant separation axioms. Definitions Suppose that X is a topological space. X is a Hausdorff spaces, and in model theoryIn mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. It assumes that there are some pre-existing mathematical objects out there, an in the construction of ultraproductModel theory An ultraproduct is a mathematical construction, which is used in abstract algebra to construct new fields from given ones, and in model theory, a branch of mathematical logic. In particular, it can be used in a "purely semantic" proof of thes and ultrapowerAn ultrapower is an important special case of the ultraproduct construction. Suppose κ is an infinite cardinal number and F is a structure in a first-order theory; for example F could be a field. A filter (mathematics) U on κ is a subset of ths. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on posets are most important if the poset is a Boolean algebra, since in this case the ultrafilters coincide with the prime filters. Ultrafilters in this form play a central role in Stone's representation theorem for Boolean algebrasIn mathematics, Stone's representation theorem for Boolean algebras named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Boolean spaces i. totally disconnected compact Hausdorff topological space.
The set G of all ultrafilters of a poset P can be topologized in a natural way, that is in fact closely related to the abovementioned representation theorem. For any element a of P, let Da = { U in G | a in U }. This is most useful when P is again a Boolean algebra, since in this situation the set of all Da is a base for a compact Hausdorff topology on G. Especially, when considering the ultrafilters on a set S (i.e. the case that P is the powerset of S ordered via subset inclusion), the resulting topological space is the Stone-Cech compactification of a discrete space of cardinality |S|.
Ultrafilters on sets are used in the ultrapower construction of certain fields of hyperreal numbers.
Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter. Order theory| Politics | Business | Finance |
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