===== Monoid =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $M$ ... set |
| @#FFBB00: definiendum | @#FFBB00: $ \langle\!\langle M,*\rangle\!\rangle \in$ it |
| @#AAFFAA: inclusion   | @#AAFFAA: $*$ ... binary operation |
| @#FFFDDD: exists      | @#FFFDDD: $e$ |
| @#55EE55: postulate   | @#55EE55: $e$ ... unit element $\langle\!\langle M,*\rangle\!\rangle$ |
| @#55EE55: postulate   | @#55EE55: $(a*b)*c=a*(b*c)$ |

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=== 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: [[http://en.wikipedia.org/wiki/Monoid|Monoid]]

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=== Requirements ===
[[Unit element]]
=== Subset of ===
[[Semigroup]]