Inverse property loop

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 loop or inverse property loop or IP-loop if, for there exist maps $a\mapsto a^{\lambda }$ and $a\mapsto a^{\rho }$ such that:

$a^{\lambda }*(a*b)=b\mathrm{\forall }a,b\in L$

and:

$(a*b)*b^{\rho }=a\mathrm{\forall }a,b\in L$