associativ

Associativ, in mathematics and related fields, is an adjective describing the associative property of a binary operation. An operation * on a set S is associative if for all a, b, c in S, (a*b)*c = a*(b*c).

Common examples are addition and multiplication of real numbers, where grouping does not affect the result: (a+b)+c = a+(b+c). Subtraction and division are not associative: (6-2)-1 = 3, but 6-(2-1) = 5. Exponentiation is also not associative: (2^3)^2 = 64, while 2^(3^2) = 512. Function composition is associative: f∘(g∘h) = (f∘g)∘h.

Associativity is a central property in algebraic structures such as semigroups, monoids, and groups; it guarantees that the result of a long chain a*b*c*d does not depend on how parentheses are placed. Matrices under multiplication, like many other binary operations, are typically associative, so (AB)C = A(BC). Logical operations such as AND and OR are also associative in standard Boolean algebra.

Non-associative examples include subtraction, division, and, in some contexts, exponentiation when viewed abstractly. There are algebraic systems that intentionally do not satisfy associativity, such as certain non-associative algebras, including the octonions.

Understanding associativity helps in simplifying expressions and in defining well-behaved algebraic structures. It is also a consideration in computer science and symbolic computation, where parsing of expressions often relies on associative properties.