assosiative

Associative describes a property of a binary operation on a set. If you perform the operation on three elements a, b, and c, the result is the same regardless of how you group them. More formally, a binary operation * on a set S is associative if (a * b) * c = a * (b * c) for all a, b, c in S. The standard term is associative; assosiative is a common misspelling.

Common examples include addition and multiplication of numbers, and concatenation of strings. For numbers, (1 + 2) + 3 = 1 + (2 + 3) and (2 × 3) × 4 = 2 × (3 × 4). For strings, ("ab" + "c") equals "a" + ("bc"). Set theory also provides associative operations, such as union and intersection: (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C). Function composition is another example: (f ∘ g) ∘ h = f ∘ (g ∘ h).

In algebra, associativity is central: a semigroup is a set with an associative binary operation; a monoid adds an identity element; a group adds inverses. In computing, associativity allows rewriting expressions and parallel evaluation without changing the outcome.

Not all operations are associative. Subtraction, division, and general exponentiation do not satisfy the associative law in general; for example, (5 − 3) − 2 ≠ 5 − (3 − 2). Understanding associativity helps in simplifying expressions and reasoning about algebraic structures.