Semigroup/Examples/Order 2

Examples of 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}$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $1 \ \text{(ii)}$