Magma (algebra)

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For other uses, see Magma (disambiguation).

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 × MM. The binary operation must be closed by definition but no other properties are imposed.

History and terminology[edit]

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[edit]

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

a, bM: abM.

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[edit]

A morphism of magmas is a function, f : MN, mapping magma, M, to magma, N, that preserves the binary operation:

f (xM y) = f(x) •N f(y)

where •M and •N denote the binary operation on M and N respectively.

Notation and combinatorics[edit]

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.

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)).

The number of non-isomorphic magmas having 0, 1, 2, 3, 4, ... elements are 1, 1, 10, 3330, 178981952, ... (sequence A001329 in OEIS). The corresponding numbers of non-isomorphic and non-antiisomorphic magmas are 1, 1, 7, 1734, 89521056, ... (sequence A001424 in OEIS).[7]

Free magma[edit]

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:[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 : 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 magmas[edit]

Magma to group2.svg

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 non-empty associative quasigroups
Abelian groups
Groups where the operation is commutative

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

Classification by properties[edit]

Group-like structures
Totalityα Associativity Identity Divisibility Commutativity
Semicategory Unneeded Required Unneeded Unneeded Unneeded
Category Unneeded Required Required Unneeded Unneeded
Groupoid Unneeded Required Required Required Unneeded
Magma Required Unneeded Unneeded Unneeded Unneeded
Quasigroup Required Unneeded Unneeded Required Unneeded
Loop Required Unneeded Required Required Unneeded
Semigroup Required Required Unneeded Unneeded Unneeded
Monoid Required Required Required Unneeded Unneeded
Group Required Required Required Required Unneeded
Abelian Group Required Required Required Required Required
Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

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[9]
Alternative
If it satisfies the identities xxyxxy and xyyxyy
Power-associative
If the submagma generated by any element is associative
A semigroup, or associative
If it satisfies the identity, xyzxyz
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, 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, xxy, holds
A semigroup with right zeros
If it is a semigroup and, for all x, the identity, xyx, 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[edit]

See n-ary group.

See also[edit]

References[edit]

  1. ^ Hausmann, B. A.; Ore, Øystein (October 1937), "Theory of quasi-groups", American Journal of Mathematics 59 (4): 983–1004, JSTOR 2371362 
  2. ^ Hollings, Christopher (2014), Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, American Mathematical Society, pp. 142–3, ISBN 978-1-4704-1493-1 
  3. ^ 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 
  4. ^ Bourbaki, N. (1998) [1970], "Alebraic Structures: §1.1 Laws of Composition: Definition 1", Algebra I: Chapters 1–3, Springer, p. 1, ISBN 978-3-540-64243-5 
  5. ^ 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 
  6. ^ Evseev, A. E. (1988), "A survey of partial groupoids", in Silver, Ben, Nineteen Papers on Algebraic Semigroups, American Mathematical Society, ISBN 0-8218-3115-1 
  7. ^ Weisstein, Eric W., "Groupoid", MathWorld.
  8. ^ 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 
  9. ^ Kepka, T.; Němec, P. (1996), "Simple balanced groupoids" (PDF), Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 35 (1): 53–60 
  10. ^ Ježek, Jaroslav; Kepka, Tomáš (1981), "Free entropic groupoids" (PDF), Commentationes Mathematicae Universitatis Carolinae 22 (2): 223–233, MR 620359 .

Further reading[edit]