# Definition:Algebra Loop

*This page is about Algebra Loop. For other uses, see 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:

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**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.8$