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 closure (binary operation)|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 operationCommonly studied types of magmas include
- quasigroups—nonempty magmas where division is always possible;
- loop—quasigroups with identity elements;
- semigroups—magmas where the operation is associative;
- monoids—semigroups with identity elements;
- group—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 full 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 (ie (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