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 $M \times M \rightarrow M$. 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 (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 in 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 "$\cdot$" that sends any two elements $a,b \in M$ to another element $a \cdot b$. The symbol "$\cdot$" is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation $(M,\cdot)$ must satisfy the following requirement (known as the magma axiom):

For all $a, b$ in $M$, the result of the operation $a \cdot b$ is also in $M$.

And in mathematical notation:

$\forall a,b \in M : a \cdot b \in M$

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

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

Morphism of magmas

A morphism of magmas is a function $f\colon M\to N$ 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 xyz abbreviates (xy) • z. For more complex expressions the use of parentheses is reduced rather than eliminated, as in (a(bc))d = (abc)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 $f\colon X\to N$ is a function from the set $X$ to any magma $N$, then there is a unique extension of $f$ to a morphism of magmas $f^\prime$

$f^\prime\colon M_X \to N.$

Classification by properties

 Totality* Associativity Identity Divisibility Group-like structures. The entries say whether the property is required. No Yes No No No No Yes Yes No No No Yes Yes Yes No Yes No No No No Yes No No Yes No Yes No Yes Yes No Yes Yes No No No Yes Yes Yes No No Yes Yes Yes Yes No 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 xyuz = xuyz (i.e. (xy) • (uz) = (xu) • (yz) for all x, y, u, z in S)
• left semimedial if it satisfies the identity xxyz = xyxz
• right semimedial if it satisfies the identity yzxx = yxzx
• semimedial if it is both left and right semimedial
• left distributive if it satisfies the identity xyz = xyxz
• right distributive if it satisfies the identity yzx = yxzx
• 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 xxy = xx = yxx[7]
• alternative if it satisfies the identities xxy = xxy and xyy = xyy
• a semigroup if it satisfies the identity xyz = xyz (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.[citation needed]

If • is instead a partial operation, then S is called a partial magma[8] or more often a partial groupoid.[8][9]

See n-ary group.

Group
Monoid
Semigroup
 Magma
Associativity
Identity
Inverses

References

1. ^ B. A. Hausmann and O ore. Theory of quasi-groups", AJM 59(4) 983-1004, http://www.jstor.org/stable/2371362
2. ^ Christopher Hollings (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.
3. ^ George M. Bergman; Adam O. Hausknecht (1996). Cogroups and Co-rings in Categories of Associative Rings. American Mathematical Soc. p. 61. ISBN 978-0-8218-0495-7.
4. ^ N. Bourbaki (1989) [1970]. Algebra I: Chapters 1-3. Springer Science & Business Media. p. 1. ISBN 0-387-19373-1.
5. ^ http://mathworld.wolfram.com/Groupoid.html
6. ^ Rowen, Louis Halle (2008). Graduate Algebra: Noncommutative View. Graduate studies in mathematics. American Mathematical Society. p. 321. ISBN 0-8218-8408-5.
7. ^ [1]
8. ^ a b Folkert Müller-Hoissen, Jean Marcel Pallo, Jim Stasheff, ed. (2012). Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift. Springer Science & Business Media. p. 11. ISBN 978-3-0348-0405-9.
9. ^ Evseev, A. E. (1988). "A survey of partial groupoids". In Ben Silver. Nineteen Papers on Algebraic Semigroups. American Mathematical Soc. ISBN 0-8218-3115-1.