In abstract algebra, a magma, binar,[1] or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed.

## History and terminology

The term groupoid was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid. The term was then appropriated by B. A. Hausmann and Øystein Ore (1937)[2] 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.[3]

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]."[4] It also appears in Bourbaki's Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970.[5]

## Definition

A magma is a set M matched with an operation • that sends any two elements a, bM to another element, abM. 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 ab is also in M.

And in mathematical notation:

${\displaystyle a,b\in M\implies a\cdot b\in M.}$

If • is instead a partial operation, then (M, •) is called a partial magma[6] or, more often, a partial groupoid.[6][7]

## Morphism of magmas

A morphism of magmas is a function f : MN that maps magma (M, •) to magma (N, ∗) that preserves the binary operation:

f (xy) = f(x) ∗ f(y).

## 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 • (bc)) • 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: xyz ≡ (xy) • z. For example, the above is abbreviated to the following expression, still containing parentheses:

(abc)d.

A way to avoid completely the use of parentheses is prefix notation, in which the same expression would be written ••abcd. 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 nn2 magmas with n elements, so there are 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 the numbers of simultaneously non-isomorphic and non-antiisomorphic magmas are 1, 1, 7, 1734, 89521056, ... (sequence A001424 in the OEIS).[8]

## 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). The binary operation on MX is formed by wrapping each of the two operands in parentheses and juxtaposing them in the same order. For example:

ab = (a)(b),
a • (ab) = (a)((a)(b)),
(aa) • b = ((a)(a))(b).

MX can be described as the set of non-associative words on X with parentheses retained.[9]

It can also be viewed, 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.

A free magma has the universal property such that if f : XN is a function from X to any magma N, then there is a unique extension of f to a morphism of magmas f

f′ : MXN.

## 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:

Note that each of divisibility and invertibility imply the cancellation property.

Magmas with commutativity

## Classification by properties

Group-like structures
Closure Associative Identity Cancellation Commutative
Partial magma Unneeded Unneeded Unneeded Unneeded Unneeded
Semigroupoid Unneeded Required Unneeded Unneeded Unneeded
Small category Unneeded Required Required Unneeded Unneeded
Groupoid Unneeded Required Required Required Unneeded
Commutative Groupoid Unneeded Required Required Required Required
Magma Required Unneeded Unneeded Unneeded Unneeded
Commutative magma Required Unneeded Unneeded Unneeded Required
Quasigroup Required Unneeded Unneeded Required Unneeded
Commutative quasigroup Required Unneeded Unneeded Required Required
Associative quasigroup Required Required Unneeded Required Unneeded
Commutative-and-associative quasigroup Required Required Unneeded Required Required
Unital magma Required Unneeded Required Unneeded Unneeded
Commutative unital magma Required Unneeded Required Unneeded Required
Loop Required Unneeded Required Required Unneeded
Commutative loop Required Unneeded Required Required Required
Semigroup Required Required Unneeded Unneeded Unneeded
Commutative semigroup Required Required Unneeded Unneeded Required
Monoid Required Required Required Unneeded Unneeded
Commutative monoid Required Required Required Unneeded Required
Group Required Required Required Required Unneeded
Abelian group Required Required Required Required Required

A magma (S, •), with x, y, u, zS, is called

Medial
If it satisfies the identity xyuzxuyz
Left semimedial
If it satisfies the identity xxyzxyxz
Right semimedial
If it satisfies the identity yzxxyxzx
Semimedial
If it is both left and right semimedial
Left distributive
If it satisfies the identity xyzxyxz
Right distributive
If it satisfies the identity yzxyxzx
Autodistributive
If it is both left and right distributive
Commutative
If it satisfies the identity xyyx
Idempotent
If it satisfies the identity xxx
Unipotent
If it satisfies the identity xxyy
Zeropotent
If it satisfies the identities xxyxxyxx[10]
Alternative
If it satisfies the identities xxyxxy and xyyxyy
Power-associative
If the submagma generated by any element is associative
Flexible
if xyxxyx
Associative
If it satisfies the identity xyzxyz, called a semigroup
A left unar
If it satisfies the identity xyxz
A right unar
If it satisfies the identity yxzx
Semigroup with zero multiplication, or null semigroup
If it satisfies the identity xyuv
Unital
If it has an identity element
Left-cancellative
If, for all x, y, z, relation xy = xz implies y = z
Right-cancellative
If, for all x, y, z, relation 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 it satisfies the identity xyx
A semigroup with right zeros
If it is a semigroup and it satisfies the identity yxx
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.[11]

## Number of magmas satisfying given properties

Idempotence Commutative property Associative property Cancellation property OEIS sequence (labeled) OEIS sequence (isomorphism classes)
Unneeded Unneeded Unneeded Unneeded A002489 A001329
Required Unneeded Unneeded Unneeded A090588 A030247
Unneeded Required Unneeded Unneeded A023813 A001425
Unneeded Unneeded Required Unneeded A023814 A001423
Unneeded Unneeded Unneeded Required A002860 add a(0)=1 A057991
Required Required Unneeded Unneeded A076113 A030257
Required Unneeded Required Unneeded
Required Unneeded Unneeded Required
Unneeded Required Required Unneeded A023815 A001426
Unneeded Required Unneeded Required A057992
Unneeded Unneeded Required Required A034383 add a(0)=1 A000001 with a(0)=1 instead of 0
Required Required Required Unneeded
Required Required Unneeded Required a(n)=1 for n=0 and all odd n, a(n)=0 for all even n≥2
Required Unneeded Required Required a(0)=a(1)=1, a(n)=0 for all n≥2 a(0)=a(1)=1, a(n)=0 for all n≥2
Required Required Required Required a(0)=a(1)=1, a(n)=0 for all n≥2 a(0)=a(1)=1, a(n)=0 for all n≥2

## Category of magmas

The category of magmas, denoted Mag, is the category whose objects are magmas and whose morphisms are magma homomorphisms. The category Mag has direct products, and there is an inclusion functor: Set → Med ↪ Mag as trivial magmas, with operations given by projection x T y = y.

An important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the (constant sequence of the) endomorphism.

Because the singleton ({*}, *) is the terminal object of Mag, and because Mag is algebraic, Mag is pointed and complete.[12]

## References

1. ^ Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, CRC Press, ISBN 978-1-4398-5130-2
2. ^ Hausmann, B. A.; Ore, Øystein (October 1937), "Theory of quasi-groups", American Journal of Mathematics, 59 (4): 983–1004, doi:10.2307/2371362, JSTOR 2371362.
3. ^ Hollings, Christopher (2014), Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, American Mathematical Society, pp. 142–143, ISBN 978-1-4704-1493-1.
4. ^ Bergman, George M.; Hausknecht, Adam O. (1996), Cogroups and Co-rings in Categories of Associative Rings, American Mathematical Society, p. 61, ISBN 978-0-8218-0495-7.
5. ^ Bourbaki, N. (1998) [1970], "Algebraic Structures: §1.1 Laws of Composition: Definition 1", Algebra I: Chapters 1–3, Springer, p. 1, ISBN 978-3-540-64243-5.
6. ^ a b Müller-Hoissen, Folkert; Pallo, Jean Marcel; Stasheff, Jim, eds. (2012), Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift, Springer, p. 11, ISBN 978-3-0348-0405-9.
7. ^ Evseev, A. E. (1988), "A survey of partial groupoids", in Silver, Ben (ed.), Nineteen Papers on Algebraic Semigroups, American Mathematical Society, ISBN 0-8218-3115-1.
8. ^
9. ^ Rowen, Louis Halle (2008), "Definition 21B.1.", Graduate Algebra: Noncommutative View, Graduate Studies in Mathematics, American Mathematical Society, p. 321, ISBN 0-8218-8408-5.
10. ^ Kepka, T.; Němec, P. (1996), "Simple balanced groupoids" (PDF), Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, 35 (1): 53–60.
11. ^ Ježek, Jaroslav; Kepka, Tomáš (1981), "Free entropic groupoids" (PDF), Commentationes Mathematicae Universitatis Carolinae, 22 (2): 223–233, MR 0620359.
12. ^ Borceux, Francis; Bourn, Dominique (2004). Mal'cev, protomodular, homological and semi-abelian categories. Springer. pp. 7, 19. ISBN 1-4020-1961-0.