In mathematics, a binary operation * on a set

*S*is called

**associative**if for all

*x*,

*y*and

*z*in

*S*, (

*x**

*y*) *

*z*=

*x** (

*y**

*z*).

The most commonly known examples of associativity are addition and multiplication of natural numbers; for example:

- (7 + 3) + 9 = 7 + (3 + 9), since the expression on the left evaluates to 10 + 9 = 19, which the expression on the right evaluates to 7 + 12 = 19, the same value;
- (10 × 5) × 3 = 10 × (5 × 3), since the expression on the left evaluates to 50 × 3 = 150, while the expression on the right evalutes to 10 × 15 = 150.

*M*is some set and

*S*denotes the set of all functions from

*M*to

*M*, then the operation of functional composition on

*S*is associative.

On the other hand, exponentiation is *not* associative; eg 2^(1^3)=2 but (2^1)^3=8. But the usual superscript notation tends to prevent questions of order arising.

A set with an associative binary operation on it is called a semigroup; monoids and groupss are examples of semigroups.

See also Commutativity, Distributive property, Identity element\n