additivassoziativ

Additivassoziativität is not a standard standalone term in most mathematical texts. It is sometimes used informally to describe a compatibility between two binary operations on a set: an addition operation and a second operation that behaves like a product. In this sense, a structure with a set A equipped with an addition operation + and another binary operation ◦ is said to be additivassoziativ when the following conditions hold in a compatible way: + is associative (and typically commutative), ◦ is associative, and the distributive laws hold with respect to + in both arguments: a◦(b+c) = a◦b + a◦c and (a+b)◦c = a◦c + b◦c for all a,b,c in A. Under these conditions ◦ is bilinear with respect to +, and the whole structure resembles a ring or algebra.

In standard terminology, the properties described by additivassoziativität are usually separated and denoted as additivity, associativity, and distributivity. The combination of an abelian group structure under + with an associative, distributive operation ◦ is precisely the core pattern of ring theory and algebraic structures such as algebras and modules. Thus, while additivassoziativität may name a familiar compatibility, mathematicians typically refer to bilinearity, distributivity, associativity, and the ring-like or algebraic framework instead.

Examples often cited include the real numbers, complex numbers, or matrices with ordinary addition and multiplication: + is associative (and commutative in many cases), ◦ (multiplication) is associative and distributes over +. These standard structures illustrate the familiar interplay between additive and multiplicative-like operations that underpins many areas of algebra.

See also: Additivity, Associativity, Bilinear map, Ring, Algebra, Distributivity.