| context | $X$ |
| postulate | $ \langle X,* \rangle \in \text{Loop}(X)$ |
| context | $\langle X,* \rangle \in \mathrm{Quasigroup}(X)$ |
| range | $e,a\in X$ |
| postulate | $\exists e.\ \forall a.\ (a*e=e*a=a) $ |
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 loop, i.e. the set where the operation “$*$” is defined on.
Wikipedia: Quasigroup