# Magma algebra

In abstract algebra, a **magma** (also called a **groupoid**) is a particularly basic kind of algebraic structure. Specifically, a magma consists of a set *M* equipped with a single binary operation *M* × *M* → *M*. A binary operation is closed by definition, but no other axioms are imposed on the operation.

The term *magma* for this kind of structure was introduced by Bourbaki; however, the term *groupoid* is a very common alternative. Unfortunately, the term *groupoid* also refers to an entirely different kind of algebraic concept described at groupoid.

## Types of magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include

- quasigroups—nonempty magmas where division is always possible;
- loops—quasigroups with identity elements;
- semigroups—magmas where the operation is associative;
- monoids—semigroups with identity elements;
- groups—monoids with inverse elements, or equivalently, associative quasigroups (which are always loops);
- abelian groups—groups where the operation is commutative.

## Free magma

A **free magma** on a set *X* is the "most general possible" magma generated by the set *X* (that is there are no relations or axioms imposed on the generators; see free object). It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labelled by elements of *X*. The operation is that of joining trees at the root. It therefore has a foundational role in syntax.

*See also*: free semigroup, free group.

## More definitions

A magma (*S*, *) is called

*medial*if it satisfies the identity*xy***uz*=*xu***yz*(i.e. (*x***y*) * (*u***z*) = (*x***u*) * (*y***z*) for all*x*,*y*,*u*,*z*in*S*),*left semimedial*if it satisfies the identity*xx***yz*=*xy***xz*,*right semimedial*if it satisfies the identity*yz***xx*=*yx***zx*,*semimedial*if it is both left and right semimedial,*left distributive*if it satisfies the identity*x***yz*=*xy***xz*,*right distributive*if it satisfies the identity*yz***x*=*yx***zx*,*autodistributive*if it is both left and right distributive,*commutative*if it satisfies the identity*xy*=*yx*,*idempotent*if it satisfies the identity*xx*=*x*,*unipotent*if it satisfies the identity*xx*=*yy*,*zeropotent*if it satisfies the identity*xx***y*=*yy***x*=*xx*,*alternative*if it satisfies the identities*xx***y*=*x***xy*and*x***yy*=*xy***y*,*power-associative*if the submagma generated by any element is associative,- a
*semigroup*if it satisfies the identity*x***yz*=*xy***z*(associativity), - a
*semigroup with left zeros*if it satisfies the identity*x*=*xy*, - a
*semigroup with right zeros*if it satisfies the identity*x*=*yx*, - a
*semigroup with zero multiplication*if it satisfies the identity*xy*=*uv*, - a
*left unar*if it satisfies the identity*xy*=*xz*, - a
*right unar*if it satisfies the identity*yx*=*zx*, *trimedial*if any triple of its (not necessarily distinct) elements generates a medial submagma,*entropic*if it is a homomorphic image of a medial cancellation magma.

## See also

## External links

- Jezek page
- Definition list J.Jezek and T.Kepka: Medial groupoids Rozpravy CSAV, Rada mat. a prir. ved 93/2 (1983), 93 pp
- Definition list but old groupoid for magma
- medial groupoid groupoid = magma
- A Catalogue of Algebraic Systems / John Pedersen no broken links
- Mathematical Structures: medial groupoids groupoid = magma
- operations
- mathworld: Groupoid

cs:Grupoid de:Magma (Mathematik) et:Rühmoid es:Magma (álgebra) fr:Magma (mathématiques) it:Magma (matematica) hu:Grupoid sl:Grupoid sv:Magma (matematik)