# 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).

## 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.