Kommutative

The term kommutative (commutative) describes a property of a binary operation in which changing the order of the operands does not change the result. Formally, an operation ∘ on a set S is commutative if a ∘ b = b ∘ a for all a, b in S. This idea is central in many areas of algebra and its applications.

Common examples are the addition and multiplication of numbers: for any integers a and b, a + b = b + a and a · b = b · a. In contrast, subtraction and division are not commutative: a − b ≠ b − a in general, and a ÷ b ≠ b ÷ a.

In abstract algebra, commutativity is used to define special classes of algebraic structures. A commutative semigroup is a semigroup whose operation is commutative; if there is an identity element, it is a commutative monoid; if every element has an inverse, a commutative group (also called an abelian group). A ring is called commutative when its multiplication is commutative. Not all natural operations are commutative; for example, matrix multiplication is generally non-commutative, though some special matrices do commute.

The property supports many simplifications and theoretical results. It underpins polynomial arithmetic, factorization, and many areas of mathematics and physics where the order of combining quantities is irrelevant. Understanding whether a given operation is commutative helps determine appropriate algebraic methods and the structure of the objects being studied.