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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 Halpern–Läuchli theorem ： ウィキペディア英語版 Halpern–Läuchli theorem In mathematics, the Halpern–Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false. It is often called the Halpern–Läuchli theorem, but the proper attribution for the theorem as it is formulated below is to Halpern–Läuchli–Laver–Pincus or HLLP (named after James D. Halpern, Hans Läuchli, Richard Laver, and David Pincus), following (Milliken 1979). Let d,r < ω, $\backslash langle\; T\_i:\; i\; \backslash in\; d\; \backslash rangle$ be a sequence of finitely splitting trees of height ω. Let :$\backslash bigcup\_\; \backslash left(\backslash prod\_T\_i(n)\backslash right)\; =\; C\_1\; \backslash cup\; \backslash cdots\; \backslash cup\; C\_r,$ then there exists a sequence of subtrees $\backslash langle\; S\_i:\; i\; \backslash in\; d\; \backslash rangle$ strongly embedded in $\backslash langle\; T\_i:\; i\; \backslash in\; d\; \backslash rangle$ such that :$\backslash bigcup\_\; \backslash left(\backslash prod\_S\_i(n)\backslash right)\; \backslash subset\; C\_k\backslash textk\; \backslash le\; r.$ Alternatively, let : $S^d\_\; =\; \backslash bigcup\_\; \backslash left(\backslash prod\_T\_i(n)\backslash right)$ and : $\backslash mathbb^d=\backslash bigcup\_\; S^d\_.$. The HLLP theorem says that not only is the collection $\backslash mathbb^d$ partition regular for each ''d'' < ''ω'', but that the homogeneous subtree guaranteed by the theorem is strongly embedded in :$T=\; \backslash langle\; T\_i:\; i\; \backslash in\; d\; \backslash rangle.\backslash$ ==References== #J.D. Halpern and H. Läuchli, A partition theorem, ''Trans. Amer. Math. Soc.'' 124 (1966), 360–367 #Keith R. Milliken, A Ramsey Theorem for Trees, ''J. Comb. Theory (Series A)'' 26 (1979), 215–237 #Keith R. Milliken, A Partition Theorem for the Infinite Subtrees of a Tree, ''Trans. Amer. Math. Soc.'' 263 No.1 (1981), 137–148 #J.D. Halpern and David Pincus, Partitions of Products, ''Trans. Amer. Math. Soc.'' 267, No.2 (1981), 549–568. 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「Halpern–Läuchli 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.