# Definition:Magma

## Definition

A **magma** is an algebraic structure $\struct {S, \circ}$ such that $S$ is closed under $\circ$.

That is, a **magma** is a pair $\struct {S, \circ}$ where:

- $S$ is a set
- $\circ : S \times S \to S$ is a binary operation on $S$

## Also known as

The word **magma** is a recently-coined term, and as such has not yet filtered into the mainstream literature.

Thus a **magma** is frequently referred to by description, as a **closed algebraic structure**.

Another older term for this concept is **groupoid** (or **gruppoid**). This word was first coined by Øystein Ore.

The term **groupoid** is often used for a completely different concept in category theory.

The word **groupoid** arises as a back-formation from **group** in the same way as **humanoid** derives from **human**.

The word **gruppoid** (rarely found in English) is the German term (from the German **gruppe** for **group**).

Some sources use the term **monoid**, in particular when its operation is commutative, but this is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$, as **monoid** is used for something else.

## Also defined as

Note that as usually defined, $\O \subseteq S$, that is, the underlying set is allowed (in the extreme case) to be the empty set.

However, some treatments insist that $S \ne \O$.

It may be necessary to check which definition is being referred to in any given context.

## Examples

### Real Numbers

Let $\R$ be the set of real numbers.

$\struct{\R, +}$, $\struct{\R, -}$ and $\struct{\R, \times}$ are all magmas.

## Also see

- Results about
**magmas**can be found**here**.

## Linguistic Note

The term **magma** was coined by Bourbaki.

The word has several meanings in French, but its interpretation as **jumble** is the one which was probably originally intended.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$ - 1981: Stanley Burris and H.P. Sankappanavar:
*A Course in Universal Algebra*: $\text {II} \ \S 1$