commutatif

Commutativity is a property of certain binary operations on a set, saying that changing the order of the operands does not affect the result. An operation is commutative if a ◦ b = b ◦ a for all elements a and b in the set. Structures in which this holds are often described as commutative, and in algebra the term is closely linked to Abelian structures.

Common examples include addition and multiplication of ordinary numbers, as well as set operations such as union and intersection, all of which satisfy a + b = b + a, a × b = b × a, a ∪ b = b ∪ a, and a ∩ b = b ∩ a. By contrast, many operations are not commutative in general; matrix multiplication and function composition, for instance, typically satisfy AB ≠ BA and f ∘ g ≠ g ∘ f.

In algebra, commutativity is a key axiom that distinguishes different structures. An Abelian group is a group with a commutative operation; a commutative ring has a commutative multiplication. Non-commutative analogues include non-Abelian groups and non-commutative rings, where the product ab need not equal ba.

Etymology: the term commutative derives from Latin commutativus, via French commutatif, and is related to the verb mutare, to change or exchange. The concept is widely used in mathematics and related fields, influencing both theoretical developments and practical computations.

See also: Abelian group, commutative algebra, commutative ring, non-commutativity, associativity, binary operation.