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 equipped with a single binary operation . The binary operation must be closed by definition but no other properties are imposed.
A magma is a set matched with an operation "" that sends any two elements to another element . The symbol "" is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation must satisfy the following requirement (known as the magma axiom):
- For all , in , the result of the operation is also in .
And in mathematical notation:
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 mapping magma to magma , that preserves the binary operation:
where and denote the binary operation on and 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 applications of the magma operator is given by the Catalan number . Thus, for example, , which is just the statement that and are the only two ways of pairing three elements of a magma with two operations. Less trivially, : , , , , and .
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 abbreviates . Further abbreviations are possible by inserting spaces, for example by writing in place of . 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.
A free magma on a set is the "most general possible" magma generated by the set (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 . 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 is a function from the set to any magma , then there is a unique extension of to a morphism of magmas
Classification by properties
|*Closure, which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently.|
|**Each element of a loop has a left and right inverse, but these need not coincide.|
A magma (S, •) is called
- unital if it has an identity element,
- 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 identity xx • y = yy • x = xx,
- alternative if it satisfies the identities xx • y = x • xy and x • yy = xy • y,
- 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 x • yz = xy • z (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.
See n-ary group.
- Magma category
- Auto magma object
- Universal algebra
- Magma computer algebra system, named after the object of this article.
- An example of a commutative non-associative magma
- Algebraic structures whose axioms are all identities
- Groupoid algebra
- Rowen, Louis Halle (2008). Graduate Algebra: Noncommutative View. Graduate studies in mathematics. American Mathematical Society. p. 321. ISBN 0-8218-8408-5.