Algebra Loop/Examples/Order 5

Example of Algebra Loop
The following is the Cayley table of an operation $\circ$ on $S = \set {e, a, b, c, d}$ such that $\struct {S, \circ}$ is an algebra loop whose identity is $e$, but which is not actually a group:


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

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

Proof
We note that:

demonstrating that $\circ$ is not associative.

Hence $\struct {S, \circ}$ is not a group.