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

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 $a, b \in L$ .

Note that for a quasigroup, the existence of both left and right inverses does not guarantee the existence of two-sided inverses.

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

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

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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:
-# '''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 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>.