Count of Commutative Binary Operations on Set/Examples/Order 2

Example of Use of Count of Binary Operations on Set
The Cayley tables for the complete set of commutative magmas 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}$


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