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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 Kuratowski's free set theorem ： ウィキペディア英語版 Kuratowski's free set theorem Kuratowski's free set theorem, named after Kazimierz Kuratowski, is a result of set theory, an area of mathematics. It is a result which has been largely forgotten for almost 50 years, but has been applied recently in solving several lattice theory problems, such as the congruence lattice problem. Denote by $()^$ the set of all finite subsets of a set $X$. Likewise, for a positive integer $n$, denote by $()^n$ the set of all $n$-elements subsets of $X$. For a mapping $\backslash Phi\backslash colon()^n\backslash to()^$, we say that a subset $U$ of $X$ is ''free'' (with respect to $\backslash Phi$), if for any $n$-element subset $V$ of $U$ and any $u\backslash in\; U\backslash setminus\; V$, $u\backslash notin\backslash Phi(V)$,. Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form $\backslash aleph\_n$. The theorem states the following. Let $n$ be a positive integer and let $X$ be a set. Then the cardinality of $X$ is greater than or equal to $\backslash aleph\_n$ if and only if for every mapping $\backslash Phi$ from $()^n$ to $()^$, there exists an $(n+1)$-element free subset of $X$ with respect to $\backslash Phi$. For $n=1$, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem. == References == * P. Erdős, A. Hajnal, A. Máté, R. Rado: ''Combinatorial Set Theory: Partition Relations for Cardinals'', North-Holland, 1984, pp. 282–285. * C. Kuratowski, ''Sur une caractérisation des alephs'', Fund. Math. 38 (1951), 14–17. * John C. Simms (1991) "Sierpiński's theorem", Simon Stevin 65: 69–163. 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「Kuratowski's free set theorem」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.