Associative property

A binary operation is associative when regrouping its operands leaves the result unchanged, as in (a + b) + c = a + (b + c).

Binary operation: A rule that assigns an output to each ordered pair of elements in a set. Associativity asks whether regrouping inputs to a binary operation preserves its output.
Addition: The arithmetic operation that combines quantities into a sum. Integer, rational, and real addition are associative, enabling regrouping of sums.
Commutative property: A property of a binary operation in which swapping two operands leaves the result unchanged. Associativity changes grouping; commutativity changes operand order.
Semigroup: A set equipped with an associative binary operation. Associativity alone, together with closure, defines the central law of a semigroup.
Set: A collection of distinct objects considered as a whole. An operation is associative relative to the set on which it is defined.
Multiplication: The arithmetic operation that combines quantities into a product. Associativity allows factors to be regrouped when evaluating products.
Identity element: An element that leaves every element unchanged when combined with it by an operation. An associative operation with an identity has a particularly useful algebraic structure.
Monoid: A semigroup with an identity element. Adding an identity to an associative operation yields a monoid.
Function: A rule assigning exactly one output to each input in its domain. A binary operation is a function whose inputs are pairs.
Matrix multiplication: A multiplication of compatible matrices defined by row-by-column sums. Matrix multiplication is associative even though it is generally not commutative.

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