Magma (algebra)

Topics in mathematics related to structure

In abstract algebra, a magma is a particularly basic kind of algebraic structure.

Specifically, a magma consists of a set X with a single binary operation on it. It is usually (but not always) interpreted as a kind of multiplication. No axioms are imposed on the operation in defining a magma. 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;
loopss -- quasigroups with identity elements;
semigroups -- magmas where the operation is associative;
monoids -- semigroups with identity elements;
groupss -- monoids with inverse elements, or equivalently, associative quasigroups (which are always loops);
abelian groups -- groups where the operation is commutative.

The term "magma" was introduced by Bourbaki. Previously, the term "groupoid" was common, and it is sometimes still used. In this encyclopedia, however, we reserve "groupoid" for a different algebraic concept, described at Groupoid.

There is such a thing as a free magma on any set X. It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labelled by elements of X, with operation the joining of trees at the root. It therefore has a foundational role in syntax.

Table of contents

1 More Definitions
2 See also
3 External links

More Definitions

A magma is called

medial if it satisfies the identity xy.uz=xu.yz (i.e. (x*y)*(u*z)=(x*u)*(y*z)),
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 xy=yx,
idempotent if it satisfies xx=x,
unipotent if it satisfies xx=yy,
zeropotent if it satisfies xx.y=yy.x=xx,
alternative if it satisfies 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 x.yz=xy.z (associativity),
a semigroup with left zeros if it satisfies x=xy,
a semigroup with right zeros if it satisfies x=yx,
a semigroup with zero multiplication if it satisfies xy=uv,
a left unar if it satisfies xy=xz,
a right unar if it satisfies 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.

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