# 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