assocative

Assocative is a common misspelling of associative, a term used to describe a fundamental property of binary operations in mathematics and computer science. The associative property states that the way in which operands are grouped does not affect the outcome of the operation when applied sequentially.

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). When this holds, the structure is often called a semigroup; if an identity element exists, it becomes a monoid; if every element has an inverse, it becomes a group.

Examples include addition and multiplication of integers or real numbers, where (a+b)+c = a+(b+c) and (a*b)*c = a*(b*c). Function composition (f∘g)∘h = f∘(g∘h), and string concatenation, also satisfy associativity. Matrix multiplication is associative as well.

Not all operations are associative. Subtraction, division, and exponentiation are usually non-associative; for instance, (5-3)-1 ≠ 5-(3-1). In boolean logic, AND and OR are associative, while some specialized operations may fail to be.

In computing, associativity influences evaluation order and optimization. Some languages enforce left- or right-associativity for operators, and parentheses can be used to override default groupings.