Monoid

A Monoid is a Semigroup with an identity element.

Def A set $S$ equipped with a binary operation is called a Monide if:

1. (clousure) $\forall s_1,s_2\in S$ $s_1{s}_2\in S$
2. (associativity) $\forall s_1,s_2,s_3$ $s_1(s_2{s}_3)=(s_1{s}_2)s_3$
3. (identity element) $\exists e\in S$ such that for evert $s\in S$ $es=se=s$

Oss The identity element is unique: suppose $e$ and $e^{\prime }$ are both identities, then $e=e{e}^{\prime }=e^{\prime }$.

Oss An element $m\in S$ is called invertible if there exists another element $m^{-1}\in S$ such that

$$m^{-1}m=m{m}^{-1}=e$$

where $m^{-1}$ is called the inverse of $m$. The inverse is unique, since $n^{-1}=n^{-1}(m{m}^{-1})=(n^{-1}m)m^{-1}=m^{-1}$.

Oss The set of invertbile elements of a monoid is a Group.

Every Semigroup $S$ can be transformed into a monoid denoted $S^1$ by adding a new element (the identity), and defining the operation with it as desired.

Examples

• the integers equipped with the standard product $(\mathbb{Z},\cdot )$ are a monoid.
• all the endofunctions of a set $X$ equipped with function composition $(End(X),\circ )$
• the set of square matrice of any field $M_n(\mathbb{F})$ equipped with the usual matrix prodcut.