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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cardinal invariant ： ウィキペディア英語版 Cardinal function In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. == Cardinal functions in set theory == * The most frequently used cardinal function is a function which assigns to a set "A" its cardinality, denoted by | ''A'' |. * Aleph numbers and beth numbers can both be seen as cardinal functions defined on ordinal numbers. * Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers. * Cardinal characteristics of a (proper) ideal ''I'' of subsets of ''X'' are: :$(I)=\backslash min\backslash \backslash subseteq\; I\; \backslash wedge\; \backslash bigcup\backslash notin\; I\backslash big\backslash \}.$ ::The "additivity" of ''I'' is the smallest number of sets from ''I'' whose union is not in ''I'' any more. As any ideal is closed under finite unions, this number is always at least $\backslash aleph\_0$; if ''I'' is a σ-ideal, then $\backslash operatorname(I)\; \backslash ge\; \backslash aleph\_1.$ :$\backslash operatorname(I)=\backslash min\backslash \backslash subseteq\; I\; \backslash wedge\backslash bigcup=X\backslash big\backslash \}.$ :: The "covering number" of ''I'' is the smallest number of sets from ''I'' whose union is all of ''X''. As ''X'' itself is not in ''I'', we must have add(''I'') ≤ cov(''I''). :$\backslash operatorname(I)=\backslash min\backslash ,$ :: The "uniformity number" of ''I'' (sometimes also written $(I)$) is the size of the smallest set not in ''I''. Assuming ''I'' contains all singletons, add(''I'') ≤ non(''I''). :$(I)=\backslash min\backslash \backslash subseteq\; I\; \backslash wedge\; (\backslash forall\; A\backslash in\; I)(\backslash exists\; B\backslash in\; )(A\backslash subseteq\; B)\backslash big\backslash \}.$ :: The "cofinality" of ''I'' is the cofinality of the partial order (''I'', ⊆). It is easy to see that we must have non(''I'') ≤ cof(''I'') and cov(''I'') ≤ cof(''I''). :In the case that $I$ is an ideal closely related to the structure of the reals, such as the ideal of Lebesgue null sets or the ideal of meagre sets, these cardinal invariants are referred to as cardinal characteristics of the continuum. * For a preordered set $(,\backslash sqsubseteq)$ the bounding number $()$ and dominating number $()$ is defined as ::$()=\backslash min\backslash big\backslash )(\backslash exists\; y\backslash in\; Y)(y\backslash not\backslash sqsubseteq\; x)\backslash big\backslash \},$ ::$()=\backslash min\backslash big\backslash )(\backslash exists\; y\backslash in\; Y)(x\backslash sqsubseteq\; y)\backslash big\backslash \}.$ * In PCF theory the cardinal function $pp\_\backslash kappa(\backslash lambda)$ is used. 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「Cardinal function」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.