Semigroup/Examples
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Examples of Semigroups
Operation Defined as $x + y + x y$ on Positive Integers
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.
Operation Defined as $x + y - x y$ on Integers
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.
Order $2$ Semigroups
The Cayley tables for the complete set of semigroups of order $2$ are listed below.
The underlying set in all cases is $\set {a, b}$.
- $\begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & a & a \\ \end{array} \qquad \begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & a & b \\ \end{array} \qquad \begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & b & b \\ \end{array}$
- $\begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & a & b \\ \end{array} \qquad \begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & b & a \\ \end{array} \qquad \begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & b & b \\ \end{array}$
- $\begin{array}{r|rr} & a & b \\ \hline a & b & a \\ b & a & b \\ \end{array}$
- $\begin{array}{r|rr} & a & b \\ \hline a & b & b \\ b & b & b \\ \end{array}$