Algebra Loop/Examples/Order 4

Example of Algebra Loop
The following are the Cayley tables of the operations $\circ$ on $S = \set {e, a, b, c}$ such that $\struct {S, \circ}$ is an algebra loop whose identity is $e$:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & b & c & e \\ b & b & c & e & a \\ c & c & e & a & b \\ \end{array} \qquad \begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \\ \end{array} \qquad \begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & c & e & b \\ b & b & e & c & a \\ c & c & b & a & e \\ \end{array} \qquad \begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & a & e \\ c & c & b & e & a \\ \end{array}$

The first two of these are the Cayley tables of:
 * the cyclic group of order $4$
 * the Klein $4$-group

while the $3$rd and $4$th are also isomorphic to the cyclic group of order $4$.

Proof
The initial specification allows us to populate the first few elements of the Cayley table:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a &  &   & \\ b & b &  &   & \\ c & c &  &   & \\ \end{array}$

Let us consider $a \circ a$.

This cannot be $a$ as there is already an $a$ in the row and column.


 * Let $a \circ a = e$:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e &  & \\ b & b &  &   & \\ c & c &  &   & \\ \end{array}$

This immediately forces:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c &  & \\ c & c & b &  & \\ \end{array}$

There are two ways to complete this.

Either $b \circ b = e$ which gives us the Klein $4$-group:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \\ \end{array}$

or $b \circ b = a$, which gives us:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & a & e \\ c & c & b & e & a \\ \end{array}$


 * Let $a \circ a = b$:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & b &  & \\ b & b &  &   & \\ c & c &  &   & \\ \end{array}$

We note that $a \circ b = e$ will not work, because that forces $a \circ c = c$ which is not allowed because there is already a $c$ in the $c$ column.

Hence we have $a \circ b = c$, which forces the completion of the cyclic group of order $4$:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & b & c & e \\ b & b & c & e & a \\ c & c & e & a & b \\ \end{array}$


 * Let $a \circ a = c$:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & c &  & \\ b & b &  &   & \\ c & c &  &   & \\ \end{array}$

This forces the completion of the following:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & c & e & b \\ b & b & e & c & a \\ c & c & b & a & e \\ \end{array}$

We then note that we can rearrange the order of the rows and columns of the remaining two tables to demonstrate that they are the cyclic group of order $4$:

Let us take:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & c & e & b \\ b & b & e & c & a \\ c & c & b & a & e \\ \end{array}$

Rearranging the order of the rows and columns as follows:


 * $\begin{array}{r|rrr}

\circ & e & b & c & a \\ \hline e & e & b & c & a \\ b & b & c & a & e \\ c & c & a & e & b \\ a & a & e & b & c \\ \end{array}$

from which the cyclic group of order $4$ is verified by inspection.

Similarly:


 * $\begin{array}{r|rrr}

\circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & a & e \\ c & c & b & e & a \\ \end{array}$

Rearranging the order of the rows and columns as follows:


 * $\begin{array}{r|rrr}

\circ & e & b & a & c \\ \hline e & e & b & a & c \\ b & b & a & c & e \\ a & a & c & e & b \\ c & c & e & b & a \\ \end{array}$

from which the cyclic group of order $4$ is also verified by inspection.