Monoid
A monoid has two properties,
- ${\bf{Associative}}: x\cdot (y\cdot z) = (x\cdot y)\cdot z$
- ${\bf{Identity / Neutral\ Element}}: \exists\ e \in G: x\cdot e\ = e\cdot x \ =\ x$
Adding a third property
- ${\bf{Inverse}}: \forall\ x \in G\ \exists \ y \in G:\ x\cdot y = y\cdot x =\ e$
Groups
A Group is a Monoid with Inverses
Abelian Group
A group with Communativity is an Abelian Group
- ${\bf{Commutativity}}: \forall x,y\in G: x\cdot y = y\cdot x$
Examples:
- $(\Z, +)$ - Abelian Group
- $(\Z, \cdot)$ - Monoid (Abelian Monoid because $n\cdot m = m \cdot n$)
- $(\Z_n, +_{mod(n)})$ = {0, 1, ..., n - 1} - Abelian Group
- $(\Z_n, \cdot_{mod(n)})$ - Monoid
- $(\R, +)$ - Abelian Group
- $(\R, \cdot)$ - Monoid
- $(M_n(\R), + )$ - Abelian Group
- $\begin{pmatrix}
2 & 0\\
1 & -1
\end{pmatrix} +
\begin{pmatrix}
3 & 1\\
0 & 1
\end{pmatrix} =
\begin{pmatrix}
5 & 1\\
1 & 0
\end{pmatrix}$
- Identity Element is the 0 matrix
- Inverse - make each element in matrix negative
- $(M_n(\R), \cdot )$ - Monoid
- $\begin{pmatrix}
2 & 0\\
1 & -1
\end{pmatrix} \cdot
\begin{pmatrix}
3 & 1\\
0 & 1
\end{pmatrix} =
\begin{pmatrix}
6 & 2\\
3 & 0
\end{pmatrix}$
- Identity Element is the identity matrix