Inverse property loop

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:

1. Existence of left and right inverses: There exist bijective maps $\lambda ,\rho :L\to L$ such that $\lambda (a)*(a*b)=(b*a)*\rho (a)=b\ \forall \ a,b\in L$.
2. 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