## Loop

### Set

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 “$*$”.

### 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 loop, i.e. the set where the operation “$*$” is defined on.

### Reference

Wikipedia: Quasigroup