Loop - Revision history

CJKG: /* Definition */

2014-07-08T12:45:18Z

Definition

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	An '''algebra loop''', sometimes simply called a '''loop''', is defined as a set <math>L</math> equipped with a binary operation <math>*</math>, such that the following are true:		An '''algebra loop''', sometimes simply called a '''loop''', is defined as a set <math>L</math> equipped with a binary operation <math>*</math>, such that the following are true:

	* [[neutral element]]: There exists an element <math>e \in L</math> such that <math>a * e = e * a = a \forall a \in l</math>. Such an <math>e</math> is termed a neutral element (or identity element or unit).		* [[neutral element]]: There exists an element <math>e \in L</math> such that <math>a * e = e * a = a</math> <math> \forall a \in L</math>. Such an <math>e</math> is termed a neutral element (or identity element or unit).
	* For any <math>a,b \in L</math> there exist unique solutions <math>x,y \in L</math> to the equation:		* For any <math>a,b \in L</math> there exist unique solutions <math>x,y \in L</math> to the equation:

Vipul at 22:06, 24 June 2012

2012-06-24T22:06:37Z

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	{{variationof\|group}}		{{variationof\|group}}
	{{magma property}}		{{magma property}}
	{{quick phrase\|[[quick phrase::group without associativity]], [[quick phrase::unital magma with unique left and right ~~quotient operations~~]]}}		{{quick phrase\|[[quick phrase::group without associativity]], [[quick phrase::unital magma with unique left and right quotients]]}}

	==Definition==		==Definition==

Vipul at 22:06, 24 June 2012

2012-06-24T22:06:15Z

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	{{variationof\|group}}		{{variationof\|group}}
	{{magma property}}		{{magma property}}
			{{quick phrase\|[[quick phrase::group without associativity]], [[quick phrase::unital magma with unique left and right quotient operations]]}}

	==Definition==		==Definition==

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	For more, refer [[:Category:Loop properties]].		For more, refer [[:Category:Loop properties]].


	===Weaker structures===		===Weaker structures===

Vipul at 23:40, 22 October 2010

2010-10-22T23:40:25Z

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	{{variationof\|group}}		{{variationof\|group}}
			{{magma property}}
	==Definition==		==Definition==

Vipul: /* Relation with other structures */

2010-10-22T23:40:05Z

Relation with other structures

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	===Stronger structures===		===Stronger structures===

	* [[~~Group~~]]		{\| class="sortable" border="1"
	* [[Moufang loop]]		! Structure !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure)!! Intermediate notions
	* [[Alternative loop]]		\|-
			\| [[Weaker than::group]] \|\| associative \|\| \|\| [[loop not implies group]] \|\| {{intermediate notions short\|loop\|group}}
			\|-
			\| [[Weaker than::Moufang loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|loop\|Moufang loop}}
			\|-
			\| [[Weaker than::Alternative loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|loop\|alternative loop}}
			\|}

			For more, refer [[:Category:Loop properties]].


	===Weaker structures===		===Weaker structures===

	* [[Quasigroup]]		* [[Stronger than::Quasigroup]]
	* [[Cancellative magma]]		* [[Stronger than::Cancellative magma]]

	==Groups associated to an algebra loop==		==Groups associated to an algebra loop==

Vipul: /* Groups associated to an algebra loop */

2010-10-22T23:37:07Z

Groups associated to an algebra loop

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	For an algebra loop <math>L</math>, every element <math>x \in L</math> defines a permutation <math>\rho_x</math> on <math>L</math> by <math>\rho_x(y) = y * x</math>. The group generated by all these permutations is termed the right multiplication group.		For an algebra loop <math>L</math>, every element <math>x \in L</math> defines a permutation <math>\rho_x</math> on <math>L</math> by <math>\rho_x(y) = y * x</math>. The group generated by all these permutations is termed the right multiplication group.

	Note that when <math>L</math> is a group, the mapping <math>x \mapsto \rho_x</math> defines an anti-isomorphism of groups, hence <math>L</math> is isomorphic to its ~~left~~ multiplication group. Further, the action of <math>L</math> (as the right multiplication group) on itself is the [[right regular representation]].		Note that when <math>L</math> is a group, the mapping <math>x \mapsto \rho_x</math> defines an anti-isomorphism of groups, hence <math>L</math> is isomorphic to its right multiplication group. Further, the action of <math>L</math> (as the right multiplication group) on itself is the [[right regular representation]].

	===Left inner mapping group===		===Left inner mapping group===

Vipul: moved Algebra loop to Loop

2010-03-09T15:46:41Z

moved Algebra loop to Loop

← Older revision	Revision as of 15:46, 9 March 2010
(No difference)

Vipul: 2 revisions

2008-05-07T22:49:50Z

2 revisions

← Older revision	Revision as of 22:49, 7 May 2008
(No difference)

Vipul at 08:50, 13 June 2007

2007-06-13T08:50:09Z

← Older revision		Revision as of 08:50, 13 June 2007
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	* [[Quasigroup]]		* [[Quasigroup]]
	* [[Cancellative magma]]		* [[Cancellative magma]]

			==Groups associated to an algebra loop==

			===Left multiplication group===

			For an algebra loop <math>L</math>, every element <math>x \in L</math> defines a permutation <math>\lambda_x</math> on <math>L</math> by <math>\lambda_x(y) = x * y</math>. The group generated by all these permutations is termed the left multiplication group. Note that the composite of <math>\lambda_x</math> and <math>\lambda_y</math> is ''not necessarily'' <math>\lambda_{x * y}</math> because we are not assuming associativity.

			Note that when <math>L</math> is a group, the mapping <math>x \mapsto \lambda_x</math> defines an isomorphism of groups, hence <math>L</math> is isomorphic to its left multiplication group. Further, the action of <math>L</math> (as the left multiplication group) on itself is the [[left regular representation]].

			===Right multiplication group===

			For an algebra loop <math>L</math>, every element <math>x \in L</math> defines a permutation <math>\rho_x</math> on <math>L</math> by <math>\rho_x(y) = y * x</math>. The group generated by all these permutations is termed the right multiplication group.

			Note that when <math>L</math> is a group, the mapping <math>x \mapsto \rho_x</math> defines an anti-isomorphism of groups, hence <math>L</math> is isomorphic to its left multiplication group. Further, the action of <math>L</math> (as the right multiplication group) on itself is the [[right regular representation]].

			===Left inner mapping group===

			The left inner mapping group for an algebra loop is the subgroup of the left multiplication group, comprising those elements that send the identity element to itself.

			When the algebra loop is a group, the left inner mapping group is the trivial subgroup.

			===Right inner mapping group===

			The right inner mapping group for an algebra loop is the subgroup of the right multiplication group, comprising those elements that send the identity element (of the loop) to itself.

Vipul: Started

2007-06-13T08:09:54Z

Started

New page

{{variationof|group}}

==Definition==

An '''algebra loop''', sometimes simply called a '''loop''', is defined as a set <math>L</math> equipped with a binary operation <math>*</math>, such that the following are true:

* [[neutral element]]: There exists an element <math>e \in L</math> such that <math>a * e = e * a = a \forall a \in l</math>. Such an <math>e</math> is termed a neutral element (or identity element or unit).
* For any <math>a,b \in L</math> there exist unique solutions <math>x,y \in L</math> to the equation:

<math>a * x = y * a = b</math>

Equivalently, an algebra loop is a quasigroup with a neutral element (also called identity element or unit).

==Relation with other structures==

===Stronger structures===

* [[Group]]
* [[Moufang loop]]
* [[Alternative loop]]

===Weaker structures===

* [[Quasigroup]]
* [[Cancellative magma]]