Binary operation

In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element.

More specifically, a binary operation on 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.

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

:[1][2][3] The closure property of a binary operation expresses the existence of a result for the operation given any pair of operands.

is called a partial binary operation.

For instance, division of real numbers is a partial binary operation, because one can not divide by zero:

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.

Typical examples of binary operations are the addition (

) 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

, is a binary operation which is not commutative since, in general,

{\displaystyle a-(b-c)\neq (a-b)-c}

Equation xy = yx), and is also not associative since

), a partial binary operation on the set of real or rational numbers, is not commutative or associative.

), 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

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

For example, scalar multiplication in linear algebra.

Also the dot product of two vectors maps

It depends on authors whether it is considered as a binary operation.