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


Requirements

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