Magma (algebra)

From Wikipedia, the free encyclopedia

In mathematics, particularly in abstract algebra, a magma (or groupoid) is a 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. The term groupoid is an older, but still common alternative which was introduced by Oystein Ore. Groupoid also refers to an entirely different algebraic structure described at Groupoid.

1 Types of magmas
2 Free magma
3 More definitions
4 References
5 See also
6 External links

[edit] 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.

From magma to group, via two alternative paths. Key:

M = magma, d = divisibility, a = associativity,

Q = quasigroup, S = semigroup, e = identity.

L = loop, i = inversibility, N = monoid, G = group

Note that both divisibility and inversibility imply

the existence of the cancellation property.

[edit] 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.

[edit] More definitions

A magma (S, *) is called

unital if it has an identity element,
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,
left-cancellative if for all x, y, and z, xy = xz implies y = z
right-cancellative if for all x, y, and z, yx = zx implies y = z
cancellative if it is both right-cancellative and left-cancellative
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.

[edit] References

This article does not cite its references or sources.
Please help improve this article by introducing appropriate citations. (help, get involved!) This article has been tagged since November 2006.