## Monoid

### Set

context | $M$ … set |

definiendum | $ \langle\!\langle M,*\rangle\!\rangle \in$ it |

inclusion | $*$ … binary operation |

exists | $e$ |

postulate | $e$ … unit element $\langle\!\langle M,*\rangle\!\rangle$ |

postulate | $(a*b)*c=a*(b*c)$ |

#### Discussion

The binary operation is often called *multiplication* and $e$ is called the *identity*, *identity element* or *unit*.

One generally calls $M$ the monoid, i.e. the set where the operation “$*$” is defined on, not the pair. For example, not that “A monoid is non-empty”.

Like above, one often uses infix notion for $*$.

#### Reference

Wikipedia: Monoid