associativo

Associativo is an adjective used in Italian and Spanish to describe a property of a binary operation or algebraic structure. In English, the corresponding term is associative, and the related noun is associativity. The word signals that the way in which operands are grouped does not affect the result.

Definition and meaning

For a set S with a binary operation ○, the operation is associative if, for all a, b, c in S, (a ○ b) ○ c = a ○ (b ○ c). This law allows one to omit parentheses when evaluating expressions with multiple applications of the operation, provided the operands come from the same set and the operation is well defined.

Common examples

Many familiar operations are associative: addition and multiplication of real numbers, function composition (f ∘ g) ∘ h = f ∘ (g ∘ h), and the union of sets. Matrix multiplication is also associative. By contrast, subtraction and division are not associative: (a − b) − c ≠ a − (b − c) in general, and (a ÷ b) ÷ c ≠ a ÷ (b ÷ c).

Algebraic and logical context

Associativity is foundational in abstract algebra, where a set with an associative binary operation forms a semigroup, and with an identity element forms a monoid; with inverses, a group. In category theory, composition of morphisms is strictly associative. In computer science and programming languages, operator associativity determines how expressions are parsed when explicit parentheses are absent.

See also

Associativity; binary operation; semigroup; monoid; group; function composition.