Inverse property loop - Revision history

Vipul: /* Definition */

2012-06-26T15:22:37Z

Definition

← Older revision		Revision as of 15:22, 26 June 2012
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	A [[loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:		A [[loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:

	# '''Existence of left and right inverses''': There ~~exists~~ bijective maps <math>\lambda,\rho:L \to L</math> such that <math>\lambda(a) * (a * b) = (b * a) * \rho(a) = b \ \forall \ a, b \in L</math>.		# '''Existence of left and right inverses''': There exist bijective maps <math>\lambda,\rho:L \to L</math> such that <math>\lambda(a) * (a * b) = (b * a) * \rho(a) = b \ \forall \ a, b \in L</math>.
	# '''Existence of two-sided inverses''': There exists a bijective map <math>{}^{-1}: L \to L</math> such that <math>a^{-1} * (a * b) = (b * a) * a^{-1} = b</math> for all <math>a,b \in L</math>.		# '''Existence of two-sided inverses''': There exists a bijective map <math>{}^{-1}: L \to L</math> such that <math>a^{-1} * (a * b) = (b * a) * a^{-1} = b</math> for all <math>a,b \in L</math>.

Vipul: /* Definition */

2012-06-26T15:22:10Z

Definition

← Older revision		Revision as of 15:22, 26 June 2012
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	A [[loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:		A [[loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:

	# '''Existence of left and right inverses''': There exists a bijective ~~map~~ <math>\lambda:L \to L</math> such that <math>\lambda(a) * (a * b) = b \ \forall \ a, b \in L</math>.		# '''Existence of left and right inverses''': There exists bijective maps <math>\lambda,\rho:L \to L</math> such that <math>\lambda(a) * (a * b) = (b * a) * \rho(a) = b \ \forall \ a, b \in L</math>.
	# '''Existence of two-sided inverses''': There exists a bijective map <math>{}^{-1}: L \to L</math> such that <math>a^{-1} * (a * b) = (b * a) * a^{-1} = b</math> for all <math>a,b \in L</math>.		# '''Existence of two-sided inverses''': There exists a bijective map <math>{}^{-1}: L \to L</math> such that <math>a^{-1} * (a * b) = (b * a) * a^{-1} = b</math> for all <math>a,b \in L</math>.

Vipul at 16:11, 9 March 2010

2010-03-09T16:11:00Z

← Older revision		Revision as of 16:11, 9 March 2010
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	{{~~algebra~~ loop property}}		{{loop property}}

	==Definition==		==Definition==

	An [[~~algebra~~ loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:		A [[loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:

	# '''Existence of left and right inverses''': There exists a bijective map <math>\lambda:L \to L</math> such that <math>\lambda(a) * (a * b) = b \ \forall \ a, b \in L</math>.		# '''Existence of left and right inverses''': There exists a bijective map <math>\lambda:L \to L</math> such that <math>\lambda(a) * (a * b) = b \ \forall \ a, b \in L</math>.

Vipul: /* Relation with other properties */

2010-03-05T22:59:40Z

Relation with other properties

← Older revision		Revision as of 22:59, 5 March 2010
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	\|-		\|-
	\| [[Weaker than::Automorphic inverse property loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|automorphic inverse property loop}}		\| [[Weaker than::Automorphic inverse property loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|automorphic inverse property loop}}
	\|-
	~~\| [[Weaker than::Left Bol loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|left Bol loop}}~~
	\|-
	~~\| [[Weaker than::Left Bruck loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|left Bruck loop}}~~
	\|-		\|-
	\| [[Weaker than::Moufang loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|Moufang loop}}		\| [[Weaker than::Moufang loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|Moufang loop}}

Vipul at 22:49, 5 March 2010

2010-03-05T22:49:15Z

← Older revision		Revision as of 22:49, 5 March 2010
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	An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:		An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:

	# '''Existence of left inverses''': There exists a bijective map <math>\lambda:L \to L</math> such that <math>\lambda(a) * (a * b) = b ~~\ \forall \ a, b \in L</math>.~~		# '''Existence of left and right inverses''': There exists a bijective map <math>\lambda:L \to L</math> such that <math>\lambda(a) * (a * b) = b \ \forall \ a, b \in L</math>.
	~~# '''Existence of right inverses''': There exists a bijective map <math>\rho:L \to L</math> such that <math>(a * b) * \rho(b) = a~~ \ \forall \ a,b \in L</math>.
	# '''Existence of two-sided inverses''': There exists a bijective map <math>{}^{-1}: L \to L</math> such that <math>a^{-1} * (a * b) = (b * a) * a^{-1} = b</math> for all <math>a,b \in L</math>.		# '''Existence of two-sided inverses''': There exists a bijective map <math>{}^{-1}: L \to L</math> such that <math>a^{-1} * (a * b) = (b * a) * a^{-1} = b</math> for all <math>a,b \in L</math>.

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	{{further\|[[equivalence of definitions of inverse property loop]]}}		{{further\|[[equivalence of definitions of inverse property loop]]}}

	Note that for a [[quasigroup]], ~~it is possible to have only the left-inverse property or only the right-inverse property, and even~~ the existence of both left and right inverses does not guarantee the existence of two-sided inverses.		Note that for a [[quasigroup]], the existence of both left and right inverses does not guarantee the existence of two-sided inverses.

	==Relation with other properties==		==Relation with other properties==
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	\|-		\|-
	\| [[Weaker than::Moufang loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|Moufang loop}}		\| [[Weaker than::Moufang loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|Moufang loop}}
			\|}

			===Weaker properties===

			{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::Left-inverse property loop]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Stronger than::Right-inverse property loop]] \|\| \|\| \|\| \|\|
	\|}		\|}

Vipul: /* Relation with other properties */

2010-03-05T19:29:00Z

Relation with other properties

← Older revision		Revision as of 19:29, 5 March 2010
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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::Group]] \|\| \|\| \|\| \|\|		\| [[Weaker than::Group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|group}}
	\|}

	~~===Weaker properties===~~

	~~{\| class="sortable" border="1"~~
	~~! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions~~
	\|-		\|-
	\| [[~~Stronger~~ than::~~Left-~~inverse property loop]] \|\| ~~the left-inverse map exists~~ \|\| \|\| \|\| \|\|		\| [[Weaker than::Automorphic inverse property loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|automorphic inverse property loop}}
	\|-		\|-
	\| [[~~Stronger~~ than::~~Right~~-inverse property loop]] \|\| ~~the right-inverse map exists~~ \|\| \|\| \|\| \|\|		\| [[Weaker than::Left Bol loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|left Bol loop}}
			\|-
			\| [[Weaker than::Left Bruck loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|left Bruck loop}}
			\|-
			\| [[Weaker than::Moufang loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|inverse property loop\|Moufang loop}}
	\|}		\|}

Vipul at 19:26, 5 March 2010

2010-03-05T19:26:53Z

← Older revision		Revision as of 19:26, 5 March 2010
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	==Definition==		==Definition==

	An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if~~, for there exist bijective maps <math>\lambda</math> and <math>\rho</math> on <math>L</math> such that~~:		An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:

	<math>\lambda(a) * (a * b) = b \ \forall \ a, b \in L</math>		# '''Existence of left inverses''': There exists a bijective map <math>\lambda:L \to L</math> such that <math>\lambda(a) * (a * b) = b \ \forall \ a, b \in L</math>.
			# '''Existence of right inverses''': There exists a bijective map <math>\rho:L \to L</math> such that <math>(a * b) * \rho(b) = a \ \forall \ a,b \in L</math>.
			# '''Existence of two-sided inverses''': There exists a bijective map <math>{}^{-1}: L \to L</math> such that <math>a^{-1} * (a * b) = (b * a) * a^{-1} = b</math> for all <math>a,b \in L</math>.

	~~and:~~		===Equivalence of definitions===

	~~<math>(a * b) * \rho(b) = a \ \forall a,b \in L</math>~~		{{further\|[[equivalence of definitions of inverse property loop]]}}

	~~The map <math>\lambda</math>~~ is ~~termed~~ the ''left-inverse ~~map''~~ and the ~~map <math>\rho</math> is termed~~ the ~~''right~~-~~inverse map''. These maps are unique~~.		Note that for a [[quasigroup]], it is possible to have only the left-inverse property or only the right-inverse property, and even the existence of both left and right inverses does not guarantee the existence of two-sided inverses.

	==Relation with other properties==		==Relation with other properties==

Vipul at 18:51, 5 March 2010

2010-03-05T18:51:41Z

← Older revision		Revision as of 18:51, 5 March 2010
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	An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if, for there exist bijective maps <math>\lambda</math> and <math>\rho</math> on <math>L</math> such that:		An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if, for there exist bijective maps <math>\lambda</math> and <math>\rho</math> on <math>L</math> such that:

	<math>~~a^{~~\lambda} * (a * b) = b \ \forall \ a, b \in L</math>		<math>\lambda(a) * (a * b) = b \ \forall \ a, b \in L</math>

	and:		and:

	<math>(a * b) * ~~b^{~~\rho} = a \ \forall a,b \in L</math>		<math>(a * b) * \rho(b) = a \ \forall a,b \in L</math>

	The map <math>\lambda</math> is termed the ''left-inverse map'' and the map <math>\rho</math> is termed the ''right-inverse map''. These maps are unique.		The map <math>\lambda</math> is termed the ''left-inverse map'' and the map <math>\rho</math> is termed the ''right-inverse map''. These maps are unique.

Vipul at 18:50, 5 March 2010

2010-03-05T18:50:17Z

@@ Line 3: / Line 3: @@
 ==Definition==
-An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse loop''' or '''inverse property loop''' or '''IP-loop''' if, for there exist maps <math>a \mapsto a^\lambda</math> and <math>a \mapsto a^\rho</math> such that:
+An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if, for there exist bijective maps <math>\lambda</math> and <math>\rho</math> on <math>L</math> such that:
 <math>a^{\lambda} * (a * b) = b \ \forall \ a, b \in L</math>
@@ Line 10: / Line 10: @@
 <math>(a * b) * b^{\rho} = a \ \forall a,b \in L</math>

Vipul: moved Inverse loop to Inverse property loop

2010-03-05T18:42:09Z

moved Inverse loop to Inverse property loop

← Older revision	Revision as of 18:42, 5 March 2010
(No difference)