# Definition:Algebra Loop

## Definition

An algebra loop $\struct {S, \circ}$ is a quasigroup with an identity element.

$\exists e \in S: \forall x \in S: x \circ e = x = e \circ x$

## Also known as

Some sources refer to an algebra loop as just a loop.

## Examples

### 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:

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 $4$.

### 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:

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

## Also see

• Results about algebra loops can be found here.