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Encyclopedia > Magma (algebra)

In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M × MM. A binary operation is closed by definition, but no other axioms are imposed on the operation. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ...


The term magma for this kind of structure was introduced by Bourbaki; however, the term groupoid is a very common alternative. Unfortunately, the term groupoid also refers to an entirely different kind of algebraic concept described at groupoid. Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ...

Contents


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

In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, a semigroup is a set with an associative binary operation on it. ... In mathematics, associativity is a property that a binary operation can have. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

Free magma

A free magma on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object). It can be described, 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. The idea of a free object in mathematics is one of the basics of abstract algebra. ... Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... In computer science, a binary tree is a tree data structure in which each node has at most two children. ... Syntax, originating from the Greek words συν (syn, meaning co- or together) and τάξις (táxis, meaning sequence, order, arrangement), can in linguistics be described as the study of the rules, or patterned relations that govern the way the words in a sentence come together. ...


See also: free semigroup, free group. In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from A, with the binary operation of concatenation. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many...


More definitions

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 it satisfies the identity x = xy,
  • a semigroup with right zeros if it satisfies the identity x = yx,
  • a semigroup with zero multiplication if it satisfies the identity xy = uv,
  • 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.

In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... For the meaning of medial in anatomy, see anatomical terms of location. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... Unipotent may mean Unipotent cell in biology Unipotent element, unipotent group, unipotent radical in mathematics This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In abstract algebra, a magma G is said to be left alternative if (xx)y=x(xy) for all x and y in G and right alternative if y(xx)=(yx)x for all x and y in G. A magma that is both left and right alternative is said... In abstract algebra, power associativity is a weak form of associativity. ... In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a*b = a*c always implies b = c. ... In mathematics, a semigroup is a set with an associative binary operation on it. ... In mathematics, associativity is a property that a binary operation can have. ... Unary numeral system, the simplest numeral system to represent natural numbers Unary operation, a kind of mathematical operator that has only one operand This is a disambiguation page: a list of articles associated with the same title. ... Unary numeral system, the simplest numeral system to represent natural numbers Unary operation, a kind of mathematical operator that has only one operand This is a disambiguation page: a list of articles associated with the same title. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a*b = a*c always implies b = c. ...

See also

In mathematics, the category of magmas (see category, magma for definitions), denoted by Mag, has as objects sets with a binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense). ... In mathematics, a magma in a category, or magma object, can be defined in a category with a cartesian product. ... Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ...

External links

  • medial groupoid groupoid = magma
  • A Catalogue of Algebraic Systems / John Pedersen no broken links
  • Mathematical Structures: medial groupoids groupoid = magma
  • operations
  • mathworld: Groupoid

  Results from FactBites:
 
Magma (algebra) - Wikipedia, the free encyclopedia (572 words)
In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure.
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.
A free magma on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object).
Magma (96 words)
Magma is molten rock often located inside a magma chamber[?] beneath the surface of the earth.
Magma in Earth is a complex high-temperature silicate solution that is ancestral to all igneous rocks, both intrusive and effusive.
Magma is under high pressure and sometimes emerges through volcanic vents in the form of lava.
  More results at FactBites »

 
 

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