Inverse property loop: Difference between revisions

Revision as of 19:29, 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 it satisfies the following equivalent conditions:

Existence of left inverses: There exists a bijective map $λ : L \to L$ such that $λ (a) * (a * b) = b \forall a, b \in L$ .
Existence of right inverses: There exists a bijective map $ρ : L \to L$ such that $(a * b) * ρ (b) = a \forall a, b \in L$ .
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, 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

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
Left Bol loop				\|FULL LIST, MORE INFO
Left Bruck loop				\|FULL LIST, MORE INFO
Moufang loop				\|FULL LIST, MORE INFO

@@ Line 22: / Line 22: @@
 ! 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}}
 |}