## 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