Editing Binary operation (section)

{{Short description|Mathematical operation with two operands}}
{{Distinguish|Bitwise operation}}

[[File:Binary operations as black box.svg|thumb|A binary operation <math>\circ</math> is a rule for combining the arguments <math>x</math> and <math>y</math> to produce <math>x\circ y</math>]]
In [[mathematics]], a '''binary operation''' or '''dyadic operation''' is a rule for combining two [[Element (mathematics)|elements]] (called [[operands]]) to produce another element. More formally, a binary operation is an [[Operation (mathematics)|operation]] of [[arity]] two.

More specifically, a '''binary operation''' on a [[Set (mathematics)|set]] is a [[binary function]] that maps every [[ordered pair|pair]] of elements of the set to an element of the set. Examples include the familiar [[arithmetic operations]] like [[addition]], [[subtraction]], [[multiplication]], set operations like union, complement, intersection. Other examples are readily found in different areas of mathematics, such as [[vector addition]], [[matrix multiplication]], and [[Conjugation (group theory)|conjugation in groups]].

A binary function that involves several sets is sometimes also called a ''binary operation''. For example, [[scalar multiplication]] of [[vector space]]s 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 [[algebraic structure|structure]]s that are studied in [[algebra]], in particular in [[semigroup]]s, [[monoid]]s, [[group (mathematics)|groups]], [[ring (algebra)|rings]], [[field (mathematics)|fields]], and [[vector space]]s.