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, M, equipped with a single binary operation, M × M → M. The binary operation must be closed by definition but no other properties are imposed.
Contents

History and terminology 1

Definition 2

Morphism of magmas 3

Notation and combinatorics 4

Free magma 5

Types of magmas 6

Classification by properties 7

Generalizations 8

See also 9

References 10

Further reading 11
History and terminology
The term groupoid was introduced in 1926 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)^{[1]} 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.^{[2]}
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 this usage because they use same word to mean "category in which all morphisms are invertible". The term magma was used by Serre [Lie Algebras and Lie Groups, 1965].”^{[3]} It also appears in Bourbaki's Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970.^{[4]}
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.
If • is instead a partial operation, then S is called a partial magma^{[5]} or more often a partial groupoid.^{[5]}^{[6]}
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, nonassociative 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))d
A 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.
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, C_{n}. Thus, for example, C_{2} = 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, C_{3} = 5: ((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), and a(b(cd)).
The number of nonisomorphic magmas having 0, 1, 2, 3, 4, ... elements are 1, 1, 10, 3330, 178981952, ... (sequence A001329 in OEIS). The corresponding numbers of nonisomorphic and nonantiisomorphic magmas are 1, 1, 7, 1734, 89521056, ... (sequence A001424 in OEIS).^{[7]}
Free magma
A free magma, M_{X}, 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 nonassociative words on X with parentheses retained:^{[8]}
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 ′ : M_{X} → N.
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

Magmas where division is always possible

Loops

Quasigroups with identity elements

Semigroups

Magmas where the operation is associative

Semilattices

Semigroups where the operation is commutative and idempotent

Monoids

Semigroups with identity elements

Groups

Monoids with inverse elements, or equivalently, associative loops or nonempty associative quasigroups

Abelian groups

Groups where the operation is commutative
Note that each of divisibility and invertibility imply the cancellation property.
Classification by properties
Grouplike structures


Totality

Associativity

Identity

Divisibility

Commutativity

Semicategory

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Category

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Groupoid

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Magma

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Quasigroup

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Loop

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Semigroup

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Monoid

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Group

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Abelian Group

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^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

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^{[9]}

Alternative

If it satisfies the identities xx • y ≡ x • xy and x • yy ≡ xy • y

Powerassociative

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

Leftcancellative

If, for all x, y, and, z, xy = xz implies y = z

Rightcancellative

If, for all x, y, and, z, yx = zx implies y = z

Cancellative

If it is both rightcancellative and leftcancellative

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.^{[10]}
Generalizations
See nary group.
See also
References
Further reading
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