Inverse property loop: Difference between revisions

Latest revision as of 15:22, 26 June 2012

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

Definition

A loop $(L, *)$ 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 exist bijective maps $λ, ρ : L \to L$ such that $λ (a) * (a * b) = (b * a) * ρ (a) = b \forall a, b \in L$ .
Existence of two-sided inverses: There exists a bijective map $- 1 : L \to L$ such that $a^{- 1} * (a * b) = (b * a) * a^{- 1} = b$ for all $a, b \in L$ .

Equivalence of definitions

Further information: equivalence of definitions of inverse property loop

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

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group				\|FULL LIST, MORE INFO
Automorphic inverse property loop				\|FULL LIST, MORE INFO
Moufang loop				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Left-inverse property loop
Right-inverse property loop

@@ Line 1: / Line 1: @@
-{{algebra loop property}}
+{{loop property}}
 ==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 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 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 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>.
@@ Line 13: / Line 12: @@
 {{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==
@@ Line 22: / Line 21: @@
 ! 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 than::Automorphic inverse property loop]] || || || || {{intermediate notions short|inverse property loop|automorphic inverse property loop}}
+|-
+| [[Weaker than::Moufang loop]] || || || || {{intermediate notions short|inverse property loop|Moufang loop}}
 |}
@@ Line 30: / Line 33: @@
 ! 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::Left-inverse property loop]] || || || ||
 |-
-| [[Stronger than::Right-inverse property loop]] || the right-inverse map exists || || || ||
+| [[Stronger than::Right-inverse property loop]] || || || ||
 |}