In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operationon a set is a binary function whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups.
A binary function that involves several sets is sometimes also called a binary operation. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar.
Binary operations are the keystone of most structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.
More precisely, a binary operation on a set is a mapping of the elements of the Cartesian product to : [1] [2] [3]
The closure property of a binary operation expresses the existence of a result for the operation given any pair of operands. [4]
If is not a function but a partial function, then is called a partial binary operation. For instance, division of real numbers is a partial binary operation, because one can not divide by zero: is undefined for every real number . In both model theory and classical universal algebra, binary operations are required to be defined on all elements of . However, partial algebras [5] generalize universal algebras to allow partial operations.
Sometimes, especially in computer science, the term binary operation is used for any binary function.
Typical examples of binary operations are the addition () and multiplication () of numbers and matrices as well as composition of functions on a single set. For instance,
Many binary operations of interest in both algebra and formal logic are commutative, satisfying for all elements and in , or associative, satisfying for all , , and in . Many also have identity elements and inverse elements.
The first three examples above are commutative and all of the above examples are associative.
On the set of real numbers , subtraction, that is, , is a binary operation which is not commutative since, in general, . It is also not associative, since, in general, ; for instance, but .
On the set of natural numbers , the binary operation exponentiation, , is not commutative since, (cf. Equation xy = yx), and is also not associative since . For instance, with , , and , , but . By changing the set to the set of integers , this binary operation becomes a partial binary operation since it is now undefined when and is any negative integer. For either set, this operation has a right identity (which is ) since for all in the set, which is not an identity (two sided identity) since in general.
Division (), a partial binary operation on the set of real or rational numbers, is not commutative or associative. Tetration (), as a binary operation on the natural numbers, is not commutative or associative and has no identity element.
Binary operations are often written using infix notation such as , , or (by juxtaposition with no symbol) rather than by functional notation of the form . Powers are usually also written without operator, but with the second argument as superscript.
Binary operations are sometimes written using prefix or (more frequently) postfix notation, both of which dispense with parentheses. They are also called, respectively, Polish notation and reverse Polish notation .
A binary operation on a set may be viewed as a ternary relation on , that is, the set of triples in for all and in .
For example, scalar multiplication in linear algebra. Here is a field and is a vector space over that field.
Also the dot product of two vectors maps to , where is a field and is a vector space over . It depends on authors whether it is considered as a binary operation.
In mathematics, the associative property is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a–1, such that a a–1 = a–1 a = 1. So, (right) division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b–1 ≠ b–1 a.
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).
In mathematics, a product is the result of multiplication, or an expression that identifies objects to be multiplied, called factors. For example, 21 is the product of 3 and 7, and is the product of and . When one factor is an integer, the product is called a multiple.
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by H, or in blackboard bold by Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form
In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as bn, where b is the base and n is the power; this is pronounced as "b (raised) to the n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:
In mathematics, an algebraic structure consists of a nonempty set A, a collection of operations on A, and a finite set of identities, known as axioms, that these operations must satisfy.
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if and are real numbers then the complex conjugate of is The complex conjugate of is often denoted as or .
In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it ; such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.
In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices.
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a ring. If the algebra is not unital, it may be made so in a standard way ; there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc.
In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
This is a glossary for the terminology in a mathematical field of functional analysis.