Magma (algebra)

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

Algebraic structures
Group-like Semigroup / Monoid Racks and quandles Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Rng Semiring Near-ring Commutative ring Domain Integral domain Field Ring theory
Lattice-like Lattice Semilattice Map of lattices Lattice theory
Module-like Module Group with operators Vector space
Algebra-like Algebra Associative Non-associative Lie algebra Graded Bialgebra

History and terminology

The term groupoid was introduced in 1926 by Heinrich Brandt describing his Brandt groupoid (the English word is a translation of 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 non 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 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 $C n$ . 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 $a (b (cd))$ .

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).^[5]

Free magma

A free magma M_X 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:^[6]

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 ′ : M_X → N.

See also: free semigroup, free group, Hall set, Wedderburn–Etherington number

Classification by properties

	Totality*	Associativity	Identity	Divisibility	Commutativity
*Group-like structures. The entries say whether the property is required.*
Semicategory	No	Yes	No	No	No
Category	No	Yes	Yes	No	No
Groupoid	No	Yes	Yes	Yes	No
Magma	Yes	No	No	No	No
Quasigroup	Yes	No	No	Yes	No
Loop	Yes	No	Yes	Yes	No
Semigroup	Yes	Yes	No	No	No
Monoid	Yes	Yes	Yes	No	No
Group	Yes	Yes	Yes	Yes	No
Abelian Group	Yes	Yes	Yes	Yes	Yes
*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 $xx • y = xx = y • xx$ ^[7]
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.^[8]

If • is instead a partial operation, then $S$ is called a partial magma^[9] or more often a partial groupoid.^[9]^[10]

Generalizations

See n-ary group.

References

↑ Hausmann, B. A.; Ore, Øystein (October 1937), "Theory of quasi-groups", American Journal of Mathematics 59 (4): 983–1004, JSTOR 2371362
↑ 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
↑ 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
↑ 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
↑ Weisstein, Eric W., "Groupoid", MathWorld.
↑ 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
↑ Kepka, T.; Němec, P. (1996), "Simple balanced groupoids" (PDF), Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 35 (1): 53–60
↑ Ježek, Jaroslav; Kepka, Tomáš (1981), "Free entropic groupoids" (PDF), Commentationes Mathematicae Universitatis Carolinae 22 (2): 223–233, MR 620359 .
↑ 9.0 9.1 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
↑ 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

M. Hazewinkel (2001), "Magma", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
M. Hazewinkel (2001), "Groupoid", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
M. Hazewinkel (2001), "Free magma", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Groupoid", MathWorld.