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.

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.

Also see

 * Definition:Unitary Magma
 * Definition:Semigroup