# 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.