## 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