# Magma (mathematics) | Wikipedia audio article

In abstract algebra, a magma (or groupoid;

not to be confused with groupoids in category theory) is a basic kind of algebraic structure.

Specifically, a magma consists of a set equipped with a single binary operation. The binary

operation must be closed by definition but no other properties are imposed.==History and terminology==

The term groupoid was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid

(translated from the German Gruppoid). The term was then appropriated by B. A. Hausmann

and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this

article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed

with this overloading of terminology. The Brandt groupoid is a groupoid in the sense

used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential

books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid

in the sense of Hausmann and Ore. Hollings (2014) writes that the term groupoid is “perhaps

most often used in modern mathematics” in the sense given to it in category theory.According

to Bergman and Hausknecht (1996): “There is no generally accepted word for a set with

a not necessarily associative binary operation. The word groupoid is used by many universal

algebraists, but workers in category theory and related areas object strongly to this

usage because they use the same word to mean ‘category in which all morphisms are invertible’.

The term magma was used by Serre [Lie Algebras and Lie Groups, 1965].” It also appears in

Bourbaki’s Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970.==Definition==

A magma is a set M matched with an operation, •, that sends any two elements a, b ∈ M

to another element, a • b. The symbol, •, is a general placeholder for a properly defined

operation. To qualify as a magma, the set and operation (M, •) must satisfy the following

requirement (known as the magma or closure axiom): For all a, b in M, the result of the operation

a • b is also in M.And in mathematical notation: a

, b

∈ M ⟹ a

⋅ b

∈ M {\displaystyle a,b\in M\implies a\cdot b\in

M} .If • is instead a partial operation, then

S is called a partial magma or more often a partial groupoid.==Morphism of magmas==

A morphism of magmas is a function, f : M → N, mapping magma M to magma N, that preserves

the binary operation: f (x •M y)=f(x) •N f(y)where •M and

•N denote the binary operation on M and N respectively.==Notation and combinatorics==

The magma operation may be applied repeatedly, and in the general, non-associative case,

the order matters, which is notated with parentheses. Also, the operation, •, is often omitted

and notated by juxtaposition: (a • (b • c)) • d=(a(bc))dA shorthand

is often used to reduce the number of parentheses, in which the innermost operations and pairs

of parentheses are omitted, being replaced just with juxtaposition, xy • z=(x • y)

• z. For example, the above is abbreviated to the following expression, still containing

parentheses: (a • bc)d.A way to avoid completely the

use of parentheses is prefix notation, in which the same expression would be written

••a•bcd. Another way, familiar to programmers, is postfix notation (Reverse Polish notation),

in which the same expression would be written abc••d•, in which the order of execution

is simply left-to-right (no Currying). The set of all possible strings consisting

of symbols denoting elements of the magma, and sets of balanced parentheses is called

the Dyck language. The total number of different ways of writing n applications of the magma

operator is given by the Catalan number, Cn. Thus, for example, C2=2, which is just the

statement that (ab)c and a(bc) are the only two ways of pairing three elements of a magma

with two operations. Less trivially, C3=5: ((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), and

a(b(cd)). There are n n 2 {\displaystyle n^{n^{2}}}

magmas with n {\displaystyle n}

elements so 1, 1, 16, 19683, 4294967296, … (sequence A002489 in the OEIS) magmas with 0, 1, 2,

3, 4, … elements. The corresponding numbers of non-isomorphic magmas are 1, 1, 10, 3330,

178981952, … (sequence A001329 in the OEIS) and of simultaneously non-isomorphic and non-antiisomorphic

magmas are 1, 1, 7, 1734, 89521056, … (sequence A001424 in the OEIS).==Free magma==

A free magma, MX, on a set, X, is the “most general possible” magma generated by X (i.e.,

there are no relations or axioms imposed on the generators; see free object). It can be

described as the set of non-associative words on X with parentheses retained.It can also

be viewed, 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. A free magma has the universal property such

that, if f : X → N is a function from X to any magma, N, then there is a unique extension

of f to a morphism of magmas, f ′ f ′ : MX → N.==Types of magma==Magmas are not often studied as such; instead

there are several different kinds of magma, depending on what axioms the operation is

required to satisfy. Commonly studied types of magma include: Quasigroup

A magma where division is always possible Loop

A quasigroup with an identity element Semigroup

A magma where the operation is associative Semilattice

A semigroup where the operation is commutative and idempotent

Monoid A semigroup with an identity element

Group A monoid with inverse elements, or equivalently,

an associative loop, or a non-empty associative quasigroup

Abelian group A group where the operation is commutativeNote

that each of divisibility and invertibility imply the cancellation property.==Classification by properties==

A magma (S, •), with x, y, u, z ∈ S, is called Medial

If it satisfies the identity, xy • uz ≡ xu • yz

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 identities, xx • y ≡ xx

≡ y • 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, or associative

If it satisfies the identity, x • yz ≡ xy • z

A left unar If it satisfies the identity, xy ≡ xz

A right unar If it satisfies the identity, yx ≡ zx

Semigroup with zero multiplication, or null semigroup

If it satisfies the identity, xy ≡ uv Unital

If it has an identity element 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 with left zeros

If it is a semigroup and, for all x, the identity, x ≡ xy, holds

A semigroup with right zeros If it is a semigroup and, for all x, the identity,

x ≡ yx, holds Trimedial

If any triple of (not necessarily distinct) elements generates a medial submagma

Entropic If it is a homomorphic image of a medial cancellation

magma.==Generalizations==

See n-ary group.==See also==

Magma category Auto magma object

Universal algebra Magma computer algebra system, named after

the object of this article. Commutative non-associative magmas

Algebraic structures whose axioms are all identities

Groupoid algebra