Semigroup/Examples

Examples of Semigroups

Let $\circ: \Z_{\ge 0} \times \Z_{\ge 0}$ be the operation defined on the integers $\Z_{\ge 0}$ as:

$\forall x, y \in \Z_{\ge 0}: x \circ y := x + y + x y$

Then $\struct {\Z_{\ge 0}, \circ}$ is a semigroup.

Let $\circ: \Z \times \Z$ be the operation defined on the integers $\Z$ as:

$\forall x, y \in \Z: x \circ y := x + y - x y$

Then $\struct {\Z, \circ}$ is a semigroup.

The Cayley tables for the complete set of semigroups of order $2$ are listed below.

The underlying set in all cases is $\set {a, b}$.

$\begin{array}{r|rr}

 & a & b \\

\hline a & a & a \\ b & a & a \\ \end{array} \qquad \begin{array}{r|rr}

 & a & b \\

\hline a & a & a \\ b & a & b \\ \end{array} \qquad \begin{array}{r|rr}

 & a & b \\

\hline a & a & a \\ b & b & b \\ \end{array}$

$\begin{array}{r|rr}

 & a & b \\

\hline a & a & b \\ b & a & b \\ \end{array} \qquad \begin{array}{r|rr}

 & a & b \\

\hline a & a & b \\ b & b & a \\ \end{array} \qquad \begin{array}{r|rr}

 & a & b \\

\hline a & a & b \\ b & b & b \\ \end{array}$

$\begin{array}{r|rr}

 & a & b \\

\hline a & b & a \\ b & a & b \\ \end{array}$

$\begin{array}{r|rr}

 & a & b \\

\hline a & b & b \\ b & b & b \\ \end{array}$