翻訳と辞書
Words near each other
・ Ultracote
・ Ultracraft Calypso
・ Ultracrepidarianism
・ UltraDefrag
・ Ultradian rhythm
・ Ultradrive
・ Ultradrug
・ UltraEdit
・ Ultraelectromagneticpop!
・ UltraESB
・ Ultrafast electron diffraction
・ Ultrafast laser spectroscopy
・ Ultrafast molecular process
・ Ultrafast monochromator
・ Ultrafast x-rays
Ultrafilter
・ Ultrafiltered milk
・ Ultrafiltration
・ Ultrafiltration (industrial)
・ Ultrafiltration (renal)
・ Ultrafine particle
・ Ultrafinitism
・ Ultraflight Lazair
・ UltraFly Model Corporation
・ Ultraforce
・ Ultrafox
・ Ultragaz
・ Ultraglide in Black
・ Ultrahazardous activity
・ Ultrahigh


Dictionary Lists
翻訳と辞書 辞書検索 [ 開発暫定版 ]
スポンサード リンク

Ultrafilter : ウィキペディア英語版
Ultrafilter

In the mathematical field of set theory, an ultrafilter is a maximal filter, that is, a filter that cannot be enlarged. Filters and ultrafilters are special subsets of partially ordered sets. Ultrafilters can also be defined on Boolean algebras and sets:
* An ultrafilter on a poset ''P'' is a maximal filter on ''P''.
* An ultrafilter on a Boolean algebra ''B'' is an ultrafilter on the poset of non-zero elements of ''B''.
* An ultrafilter on a set ''X'' is an ultrafilter on the Boolean algebra of subsets of ''X''.
Rather confusingly, an ultrafilter on a poset ''P'' or Boolean algebra ''B'' is a subset of ''P'' or ''B'', while an ultrafilter on a set ''X'' is a collection of subsets of ''X''. Ultrafilters have many applications in set theory, model theory, and topology.
An ultrafilter on a set ''X'' has some special properties. For example, given any subset ''A'' of ''X'', the ultrafilter must contain either ''A'' or its complement . In addition, an ultrafilter on a set ''X'' may be considered as a finitely additive measure. In this view, every subset of ''X'' is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0).
==Formal definition for ultrafilter on a set ==
Given a set ''X'', an ultrafilter on ''X'' is a set ''U'' consisting of subsets of ''X'' such that
#The empty set is not an element of ''U''
#If ''A'' and ''B'' are subsets of ''X'', ''A'' is a subset of ''B'', and ''A'' is an element of ''U'', then ''B'' is also an element of ''U''.
#If ''A'' and ''B'' are elements of ''U'', then so is the intersection of ''A'' and ''B''.
#If ''A'' is a subset of ''X'', then either ''A'' or ''X'' \ ''A'' is an element of ''U''. (Note: axioms 1 and 3 imply that ''A'' and cannot ''both'' be elements of ''U''.)
A characterization is given by the following theorem.
A filter ''U'' on a set ''X'' is an ultrafilter if any of the following conditions are true:
#There is no filter ''F'' finer than ''U'', i.e., U\subseteq F implies ''U'' = ''F''.
#A\cup B\in U implies A\in U or B\in U.
#\forall A\subseteq X\colon A\in U or X\setminus A \in U.
Another way of looking at ultrafilters on a set ''X'' is to define a function ''m'' on the power set of ''X'' by setting ''m''(''A'') = 1 if ''A'' is an element of ''U'' and ''m''(''A'') = 0 otherwise. Such a function is called a 2-valued morphism. Then ''m'' is a finitely additive measure on ''X'', and every property of elements of ''X'' is either true almost everywhere or false almost everywhere. Note that this does not define a measure in the usual sense, which is required to be ''countably additive''.
For a filter ''F'' that is not an ultrafilter, one would say ''m''(''A'') = 1 if ''A'' ∈ ''F'' and ''m''(''A'') = 0 if ''X'' \ ''A'' ∈ ''F'', leaving ''m'' undefined elsewhere.
A simple example of an ultrafilter is a ''principal ultrafilter'', which consists of subsets of ''X'' that contain a given element ''x'' of ''X''. All ultrafilters on a finite set are principal.
==Completeness==
The completeness of an ultrafilter ''U'' on a set is the smallest cardinal κ such that there are κ elements of ''U'' whose intersection is not in ''U''. The definition implies that the completeness of any ultrafilter is at least \aleph_0. An ultrafilter whose completeness is ''greater'' than \aleph_0—that is, the intersection of any countable collection of elements of ''U'' is still in ''U''—is called countably complete or \sigma-complete.
The completeness of a countably complete nonprincipal ultrafilter on a set is always a measurable cardinal.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Ultrafilter」の詳細全文を読む



スポンサード リンク
翻訳と辞書 : 翻訳のためのインターネットリソース

Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.