Inverse property loop: Difference between revisions

Revision as of 18:50, 5 March 2010

This article defines a property that can be evaluated for a loop.
View other properties of loops

Definition

An algebra loop $(L, *)$ is termed an inverse property loop or inverse loop or IP-loop if, for there exist bijective maps $λ$ and $ρ$ on $L$ such that:

$a^{λ} * (a * b) = b \forall a, b \in L$

and:

$(a * b) * b^{ρ} = a \forall a, b \in L$

The map $λ$ is termed the left-inverse map and the map $ρ$ is termed the right-inverse map. These maps are unique.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Left-inverse property loop	the left-inverse map exists
Right-inverse property loop	the right-inverse map exists

@@ 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>
+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.
+==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::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 || || || ||
+|-
+| [[Stronger than::Right-inverse property loop]] || the right-inverse map exists || || || ||
+|}