Count of Binary Operations on Set/Examples/Order 2/Equivalence Classes

Example of Use of Count of Binary Operations on Set
Consider the Cayley tables for the complete set of magmas of order $2$.

Let these be arranged into classes such that $2$ Cayley tables are in the same class the entry in each cell of one Cayley table is not the same as the entry in the diagonally opposite cell in the other.

The classes are as follows:


 * $(1): \quad \begin{array}{r|rr}

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


 * $(2): \quad \begin{array}{r|rr}

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


 * $(3): \quad \begin{array}{r|rr}

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


 * $(4): \quad \begin{array}{r|rr}

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


 * $(5): \quad \begin{array}{r|rr}

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


 * $(6): \quad \begin{array}{r|rr}

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


 * $(7): \quad \begin{array}{r|rr}

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


 * $(8): \quad \begin{array}{r|rr}

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


 * $(9): \quad \begin{array}{r|rr}

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


 * $(10): \quad \begin{array}{r|rr}

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

Of these:


 * $(1): \quad$ These are the constant operations on $\set {a, b}$.
 * From Constant Operation is Commutative and Constant Operation is Associative, both of these are associative and commutative.


 * $(2): \quad$ The operations in $(2)$ are both associative and commutative


 * $(3): \quad$ The operations in $(3)$ are neither associative nor commutative:


 * $(4): \quad$ The operations in $(4)$ are neither associative nor commutative:


 * $(5): \quad$ The operations in $(5)$ are both associative and commutative


 * $(6): \quad$ The operations in $(6)$ are both associative and commutative