Algebra Loop/Examples
Examples of Algebra Loops
Algebra Loops of Order $3$
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 $3$.
Algebra Loops of Order $4$
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:
while the $3$rd and $4$th are also isomorphic to the cyclic group of order $4$.
Algebra Loop of Order $5$
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:
- $\begin{array}{r|rrr}
\circ & e & a & b & c & d \\ \hline e & e & a & b & c & d \\ a & a & e & d & b & c \\ b & b & c & e & d & a \\ c & c & d & a & e & b \\ d & d & b & c & a & e \\ \end{array}$