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      • In abstract algebra, a magma, binar, 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) 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 ∈ M. 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 property):

      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 (M, •) is called a partial magma or, more often, a partial groupoid.


      Morphism of magmas


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

      f (x • y) = f(x) ∗ f(y).
      For example, with M equal to the positive real numbers and • as the geometric mean, N equal to the real number line, and ∗ as the arithmetic mean, a logarithm f is a morphism of the magma (M, •) to (N, ∗).

      proof:



      log
      ⁡


      x
      y



      =




      log
      ⁡
      x
      +
      log
      ⁡
      y

      2




      {\displaystyle \log {\sqrt {xy}}\ =\ {\frac {\log x+\log y}{2}}}


      Note that these commutative magmas are not associative; nor do they have an identity element. This morphism of magmas has been used in economics since 1863 when W. Stanley Jevons calculated the rate of inflation in 39 commodities in England in his A Serious Fall in the Value of Gold Ascertained, page 7.


      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))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. 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).


      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:

      a • b = (a)(b),
      a • (a • b) = (a)((a)(b)),
      (a • a) • b = ((a)(a))(b).
      MX 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 full binary trees with leaves labelled by elements of X. The operation is that of joining trees at the root.
      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.
      Monoid: A semigroup with an identity element.
      Group: A magma with inverse, associativity, and an identity element.
      Note that each of divisibility and invertibility imply the cancellation property.

      Magmas with commutativity

      Commutative magma: A magma with commutativity.
      Commutative monoid: A monoid with commutativity.
      Abelian group: A group with commutativity.


      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
      Flexible
      if xy • x ≡ x • yx
      Associative
      If it satisfies the identity x • yz ≡ xy • z, called a semigroup
      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, 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 xy ≡ x
      A semigroup with right zeros
      If it is a semigroup and it satisfies the identity yx ≡ x
      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.
      Central
      If it satisfies the identity xy • yz ≡ y


      Number of magmas satisfying given properties




      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.


      See also


      Magma category
      Universal algebra
      Magma computer algebra system, named after the object of this article.
      Commutative magma
      Algebraic structures whose axioms are all identities
      Groupoid algebra
      Hall set


      References




      Further reading


      Bruck, Richard Hubert (1971), A survey of binary systems (3rd ed.), Springer, ISBN 978-0-387-03497-3

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