asociativ

Asociativ, or associativity, is a property of certain binary operations on a set that allows the grouping of operands to be changed without altering the result. Formally, if a binary operation ◦ is defined on a set S, it is associative when for all a, b, c in S, (a ◦ b) ◦ c = a ◦ (b ◦ c).

Common examples include arithmetic addition and multiplication over numbers: (a + b) + c = a + (b + c) and (a · b) · c = a · (b · c). Matrix multiplication is also associative: (AB)C = A(BC), provided the products are dimensionally valid. String concatenation is another example: (xy)z = x(yz). Associativity is a fundamental feature in many mathematical and computational structures.

Non-associative operations include subtraction and division, where (a − b) − c ≠ a − (b − c) and (a ÷ b) ÷ c ≠ a ÷ (b ÷ c). Exponentiation is not generally associative either, since a^(b^c) and (a^b)^c differ in general.

In algebra, associativity defines a semigroup when a binary operation is associative. If a semigroup has an identity element, it becomes a monoid; if every element also has an inverse, it becomes a group. Associativity governs the behavior of many algebraic systems and underpins the ability to regroup terms in expressions unambiguously.

In computing and data processing, associative operations enable reordering and parallel evaluation of expressions, improving efficiency in reductions and aggregations across datasets. The concept also appears in logic, category theory, and programming language design, where well-behaved composition of operations is essential.