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	~~Hello! Let me begin by stating my title - Ron Stephenson~~. ~~Years ago we moved~~ to ~~Arizona but my spouse wants us~~ to ~~transfer~~. ~~Bookkeeping~~ is ~~what I do~~ for a ~~living~~. ~~One~~ of the ~~issues I love most~~ is ~~climbing~~ and ~~now I have time to consider~~ on ~~new issues~~.<br><br>~~Stop by my blog; [http:~~//~~www~~.~~Schwaben-gaming~~.de/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~9562 extended car warranty~~]		In [[mathematics]] and [[abstract algebra]], a '''Bol loop''' is an [[algebraic structure]] generalizing the notion of [[Group (mathematics)\|group]]. Bol loops are named for the Dutch mathematician [[Gerrit Bol]] who introduced them in {{harv\|Bol\|1937}}.

			A [[loop (algebra)\|loop]], ''L'', is said to be a '''left Bol loop''' if it satisfies the [[Identity (mathematics)\|identity]]

			:<math>a(b(ac))=(a(ba))c</math>, for every ''a'',''b'',''c'' in ''L'',

			while ''L'' is said to be a '''right Bol loop''' if it satisfies

			:<math>((ca)b)a=c((ab)a)</math>, for every ''a'',''b'',''c'' in ''L''.

			These identities can be seen as weakened forms of [[associativity]].

			A loop is both left Bol and right Bol if and only if it is a [[Moufang loop]]. Different authors use the term "Bol loop" to refer to either a left Bol or a right Bol loop.

			==Bruck loops==
			A Bol loop satisfying the ''automorphic inverse property,'' (''ab'')<sup>−1</sup> = ''a''<sup>−1</sup> ''b''<sup>−1</sup> for all ''a,b'' in ''L'', is known as a (left or right) '''Bruck loop''' or '''K-loop''' (named for the American mathematician [[Richard Bruck]]). The example in the following section is a Bruck loop.

			Bruck loops have applications in [[special relativity]]; see Ungar (2002). Left Bruck loops are equivalent to Ungar's (2002) '''gyrocommutative [[gyrogroup]]s''', even though the two structures are defined differently.

			==Example==
			Let ''L'' denote the set of ''n x n'' [[Positive-definite matrix\|positive definite]], [[Hermitian matrix\|Hermitian]] [[Matrix (mathematics)\|matrices]] over the complex numbers. It is generally not true that the [[Matrix multiplication\|matrix product]] ''AB'' of matrices ''A'', ''B'' in ''L'' is Hermitian, let alone positive definite. However, there exists a unique ''P'' in ''L'' and a unique [[unitary matrix]] ''U'' such that ''AB = PU''; this is the [[polar decomposition]] of ''AB''. Define a binary operation * on ''L'' by ''A'' * ''B'' = ''P''. Then (''L'', ) is a left Bruck loop. An explicit formula for is given by ''A'' * ''B'' = (''A B''<sup>2</sup> ''A'')<sup>1/2</sup>, where the superscript 1/2 indicates the unique positive definite Hermitian [[Square root of a matrix\|square root]].

			==References==
			*{{Citation \| last1=Bol \| first1=G. \| title=Gewebe und gruppen \| doi=10.1007/BF01594185 \| mr=1513147 \| zbl = 0016.22603 \| jfm = 63.1157.04 \| year=1937 \| journal=[[Mathematische Annalen]] \| issn=0025-5831 \| volume=114 \| issue=1 \| pages=414–431}}
			* Kiechle, H. (2002) ''Theory of K-Loops''. Springer. ISBN 978-3-540-43262-3.
			* Pflugfelder, H. O. (1990) ''Quasigroups and Loops: Introduction''. Heldermann. ISBN 978-3-88538-007-8 . Chapter VI is about Bol loops.
			* Robinson, D. A. (1966) "Bol loops," ''Trans. Amer. Math. Soc.'' '''123''': 341-354.
			* Ungar, A. A. (2002) ''Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces''. Kluwer. ISBN 978-0-7923-6909-7.

			[[Category:Non-associative algebra]]
			[[Category:Group theory]]


			{{Abstract-algebra-stub}}

78.91.53.232 at 17:43, 9 October 2011

2011-10-09T17:43:07Z

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Hello! Let me begin by stating my title - Ron Stephenson. Years ago we moved to Arizona but my spouse wants us to transfer. Bookkeeping is what I do for a living. One of the issues I love most is climbing and now I have time to consider on new issues.<br><br>Stop by my blog; [http://www.Schwaben-gaming.de/index.php?mod=users&action=view&id=9562 extended car warranty]

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78.91.53.232 at 17:43, 9 October 2011