assoziatives

Assoziatives is a term used in mathematics to refer to the concept of associativity or to objects built on associative operations. The word is often used to describe binary operations that obey the associativity law, or the structures that arise from such operations.

Definition and core idea: A binary operation is associative on a set if for all a, b, c in the set, (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c). This means that when applying the operation to three elements, the way the elements are grouped does not change the result. Associativity is a structural property rather than a property of individual elements.

Examples: Many familiar operations are associative. Addition and multiplication of numbers are both associative. Matrix multiplication is associative, as is function composition. Union (and intersection) of sets and string concatenation are also associative. These examples illustrate how grouping can be rearranged without altering the outcome.

Non-examples and clarifications: Not all operations are associative. Subtraction, division, and many forms of exponentiation are not; for instance, (5 − 3) − 2 ≠ 5 − (3 − 2). Understanding associativity helps avoid ambiguity in expressions and in designing algorithms and algebraic systems.

Algebraic structures: Associativity is central to several algebraic definitions. A semigroup is a set with an associative binary operation. A monoid adds an identity element; a group adds inverses. Rings and algebras extend these ideas with additional operations and structures, all of which rely on the associativity of the primary operation.

See also: associative property, semigroup, monoid, group, binary operation. In computing, the term “associative” also appears in reference to data structures such as associative arrays, though this usage is distinct from mathematical associativity.