| 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 “$*$”.
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.
Wikipedia: Quasigroup