Algebra Loop/Examples/Order 3

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


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

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

Hence, up to isomorphism, there is only one algebra loop with $3$ elements.

This is isomorphic to the additive group of integers modulo $3$.

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


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

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

Then we note that:

because both $a$ and $b$ already appear in the row and column of those cells.

Hence we have:


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

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

and the remaining cells are likewise forced.

The isomorphism $\phi$ to the additive group of integers modulo $3$ can be established as:

The fact that this is indeed an isomorphism follows by inspection.