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Ultrafilter
In the Mathematics, mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a Maximal element, maximal Filter (mathematics), filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P. If X is an arbitrary set, its power set (X), ordered by set inclusion, is always a Boolean algebra (structure), Boolean algebra and hence a poset, and ultrafilters on (X) are usually called X.If X happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on (X) or an (ultra)filter just on X is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter ''of'' a partial ordered set" vs. "''on'' an arbitrary set"; i.e. they write "(ultra)filter on X" to abbreviate "(ultra)filter of (X)". An ultrafilter on a set X may be considered as a finitely additive 0-1-valued measure (mathematics), measure on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ultrafilter Lemma
In the mathematical field of set theory, an ultrafilter on a set X is a ''maximal filter'' on the set X. In other words, it is a collection of subsets of X that satisfies the definition of a filter on X and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of X that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set X can also be characterized as a filter on X with the property that for every subset A of X either A or its complement X\setminus A belongs to the ultrafilter. Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set \wp(X) and the partial order is subset inclusion \,\subseteq. This article deals specifically with ultrafilters on a set and does not cover the more general notion. There are two types of ultra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ultrafilter (set Theory)
In the mathematical field of set theory, an ultrafilter on a set (mathematics), set X is a ''maximal filter'' on the set X. In other words, it is a collection of subsets of X that satisfies the definition of a filter (set theory), filter on X and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of X that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set X can also be characterized as a filter on X with the property that for every subset A of X either A or its complement X\setminus A belongs to the ultrafilter. Ultrafilters on sets are an important special instance of Ultrafilter, ultrafilters on partially ordered sets, where the partially ordered set consists of the power set \wp(X) and the partial order is subset inclusion \,\subseteq. This article deals specifically with ultrafilters on a set and does not cover the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ultrafilter On The Powerset Of A Set
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P. If X is an arbitrary set, its power set (X), ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on (X) are usually called X.If X happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on (X) or an (ultra)filter just on X is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter ''of'' a partial ordered set" vs. "''on'' an arbitrary set"; i.e. they write "(ultra)filter on X" to abbreviate "(ultra)filter of (X)". An ultrafilter on a set X may be considered as a finitely additive 0-1-valued measure on (X). In this view, every subset of X is either considered "almost everything" (has measure 1) or "almos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
