## Quasigroup

### Set

 context $X$
 postulate $\langle X,* \rangle \in \text{Quasigroup}(X)$
 context $\langle X,* \rangle \in \mathrm{Magma}(X)$
 range $a,b,x,y\in X$
 postulate $\forall a.\ \forall b.\ \exists x.\ a*x=b$ postulate $\forall a.\ \forall b.\ \exists y.\ y*a=b$

Here we used infix notation for “$*$”.

### Ramifications

#### Discussion

The binary operation is often called multiplication.

The axioms $*\in \mathrm{binaryOp}(X)$ above means that a monoid is closed with respect to the multiplication.

One generally calls $X$ the quasigroup, i.e. the set where the operation “$*$” is defined on.

### Reference

Wikipedia: Quasigroup