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Concept
Binary operation
Summary
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, an internal binary operation on a set is a binary operation 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.
An operation of arity two 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. Such binary operations may be called simply binary functions.
Binary operations are the keystone of most algebraic 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 :
Because the result of performing the operation on a pair of elements of is again an element of , the operation is called a closed (or internal) binary operation on (or sometimes expressed as having the property of closure).
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't 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 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.
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