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2013-03-15T08:29:34Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q175648

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	~~The author~~ is ~~known as Irwin~~. ~~Hiring~~ is ~~her day job now and she will not change it anytime soon~~. ~~Playing baseball~~ is the ~~pastime he will never quit performing~~. ~~Puerto Rico is exactly where he's been living for many years~~ and ~~he will never transfer~~.<br><br>~~Check out my website~~ :: ~~std home test (~~[~~http~~://~~fastrolls~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~72113/info/ this~~])		In [[mathematics]], the '''Halpern–Läuchli theorem''' is a partition result about finite products of infinite [[Tree (set theory)\|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, <math>\langle T_i: i \in d \rangle</math> be a sequence of finitely splitting trees of height ω. Let

			:<math>\bigcup_{n \in \omega} \left(\prod_{i<d}T_i(n)\right) = C_1 \cup \cdots \cup C_r,</math>

			then there exists a sequence of subtrees <math>\langle S_i: i \in d \rangle</math> [[Milliken's tree theorem\|strongly embedded]] in <math>\langle T_i: i \in d \rangle</math> such that

			:<math>\bigcup_{n \in \omega} \left(\prod_{i<d}S_i(n)\right) \subset C_k\text{ for some }k \le r. </math>

			Alternatively, let

			: <math>S^d_{\langle T_i: i \in d \rangle} = \bigcup_{n \in \omega} \left(\prod_{i<d}T_i(n)\right)</math>

			and

			: <math>\mathbb{S}^d=\bigcup_{\langle T_i: i \in d \rangle} S^d_{\langle T_i: i \in d \rangle}.</math>.

			The HLLP theorem says that not only is the collection <math>\mathbb{S}^d</math> [[partition regular]] for each ''d'' < ''ω'', but that the homogeneous subtree guaranteed by the theorem is '''strongly embedded''' in

			:<math>T= \langle T_i: i \in d \rangle.\ </math>

			==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.

			{{DEFAULTSORT:Halpern-Lauchli theorem}}
			[[Category:Ramsey theory]]
			[[Category:Theorems in the foundations of mathematics]]
			[[Category:Trees (set theory)]]

en>777sms at 03:45, 14 April 2012

2012-04-14T03:45:34Z

New page

The author is known as Irwin. Hiring is her day job now and she will not change it anytime soon. Playing baseball is the pastime he will never quit performing. Puerto Rico is exactly where he's been living for many years and he will never transfer.<br><br>Check out my website :: std home test ([http://fastrolls.com/index.php?do=/profile-72113/info/ this])

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en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q175648

en>777sms at 03:45, 14 April 2012