Magma (algebra)

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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.

The term magma for this kind of structure was introduced by Nicolas Bourbaki. The term groupoid is an older, but still commonly used alternative which was introduced by Øystein Ore.

Definition[edit]

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[edit]

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

Magma to group2.svg
Note that each of divisibility and invertibility
imply the cancellation property.

Morphism of magmas[edit]

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[edit]

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. Of course, for more complex expressions the use of parenthesis turns out to be inevitable. A way to avoid completely the use of parentheses is prefix notation.

Free magma[edit]

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:[1]

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.

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

Classification by properties[edit]

Group-like structures
Totality* Associativity Identity Divisibility Commutativity
Magma Yes No No No 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
Loop Yes No Yes Yes No
Quasigroup Yes No No Yes No
Groupoid No Yes Yes Yes No
Category No Yes Yes No No
Semicategory No Yes No No No
*Closure, which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently.

A magma (S, •) is called

  • unital if it has an identity element,
  • 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 identity xxy = yyx = xx,
  • alternative if it satisfies the identities xxy = xxy and xyy = xyy,
  • 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 if it satisfies the identity xyz = xyz (associativity),
  • a semigroup with left zeros if there are elements x for which the identity x = xy holds,
  • a semigroup with right zeros if there are elements x for which the identity x = yx holds,
  • a semigroup with zero multiplication or a null semigroup if it satisfies the identity xy = uv, for all x,y,u and v
  • a left unar if it satisfies the identity xy = xz,
  • a right unar if it satisfies the identity yx = zx,
  • trimedial if any triple of its (not necessarily distinct) elements generates a medial submagma,
  • entropic if it is a homomorphic image of a medial cancellation magma.

If • is instead a partial operation, then S is called a partial magma.

Generalizations[edit]

See n-ary group.

See also[edit]

Group
Monoid
Semigroup
Magma
Operation
Closure
Associativity
Identity
Inverses

References[edit]

  1. ^ Rowen, Louis Halle (2008). Graduate Algebra: Noncommutative View. Graduate studies in mathematics. American Mathematical Society. p. 321. ISBN 0-8218-8408-5.