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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
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.
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
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