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.
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) 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 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].” It also appears in Bourbaki's Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970.
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.
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.
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.
Combinatorics and parentheses
For the general, non-associative case, the magma operation may be repeatedly iterated. To denote pairings, parentheses are used. The resulting string consists of symbols denoting elements of the magma, and balanced sets of parenthesis. The set of all possible strings of balanced parenthesis 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)).
A shorthand is often used to reduce the number of parentheses. This is accomplished by using juxtaposition in place of the operation. For example, if the magma operation is •, then xy • z abbreviates (x • y) • z. For more complex expressions the use of parentheses is reduced rather than eliminated, as in (a(bc))d = (a • bc)d. A way to avoid completely the use of parentheses is prefix notation.
The number of nonisomorphic magmas having 1, 2, 3, 4, ... elements are 1, 10, 3330, 178981952, ... (sequence A001329 in OEIS). The corresponding numbers of nonisomorphic and nonantiisomorphic magmas are 1, 7, 1734, 89521056, ... (sequence A001424 in OEIS).
A free magma MX on a set X is the "most general possible" magma generated by the set 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 the set X to any magma N, then there is a unique extension of f to a morphism of magmas f ′
- f ′ : MX → N.
Classification by properties
|Group-like structures. The entries say whether the property is required.|
|*Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.|
A magma (S, •) is called
- medial if it satisfies the identity xy • uz = xu • yz (i.e. (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 identities xx • y = xx = y • xx
- alternative if it satisfies the identities xx • y = x • xy and x • yy = xy • y
- a semigroup if it satisfies the identity x • yz = xy • z (associativity)
- a left unar if it satisfies the identity xy = xz
- a right unar if it satisfies the identity yx = zx
- a semigroup with zero multiplication or a null semigroup if it satisfies the identity xy = uv
- unital if it has an identity element
- power-associative if the submagma generated by any element is associative,
- 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 there are elements x for which the identity x = xy holds
- a semigroup with right zeros if it is a semigroup and there are elements x for which 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.
See n-ary group.
- 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
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