commutativ

Commuttativ is a term used in some mathematical writings to denote the property of commutativity, the idea that the order of operands does not affect the outcome of an operation. In this sense, a binary operation is described as commuttativ when swapping the two inputs yields the same result.

Formally, if a binary operation ⊗ is defined on a set S, the operation is commuttativ if a ⊗ b = b ⊗ a for all a, b in S. This property is central to many familiar arithmetic and set operations, and it often simplifies both reasoning and computation.

Examples of commuttativ operations include addition and multiplication of real or integer numbers, the union and intersection of sets, and vector addition in ordinary Euclidean space. In these cases, changing the order of the operands does not change the result. Non-examples include subtraction and division of numbers, and matrix multiplication in general, where a ⊗ b ≠ b ⊗ a may hold.

In abstract algebra, a structure with a commuttativ binary operation is often described as abelian or commutative. When a set equipped with a commuttativ operation also satisfies associativity and has an identity element, it forms a commutative monoid or, with inverses, a commutative group. If the structure is a ring or algebra with a commuttativ multiplication, it is referred to as a commutative ring or commutative algebra.

Notes: the standard term in modern English mathematics is commutativity and commutative for the adjective; commuttativ appears as a less common or language-specific variant. See also commutativity, abelian groups, and commutative algebra.