Inverse property loop

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