Alternative loop

Symbol-free definition
An alternative loop is an algebra loop that is also an alternative magma.

Definition with symbols
An alternative loop is an algebra loop $$(L,*)$$ satisfying the following two conditions:


 * $$\! x * (x * y) = (x * x) * y \ \forall \ x,y \in L$$.
 * $$\! x * (y * y) = (x * y) * y \ \forall \ x,y \in L$$.

Stronger properties

 * Weaker than::Di-associative loop
 * Weaker than::Moufang loop
 * Weaker than::Group

Weaker structures

 * Stronger than::Algebra loop
 * Stronger than::Left-alternative loop
 * Stronger than::Right-alternative loop
 * Stronger than::Alternative magma
 * Stronger than::Quasigroup