# (Function|Operator) - Associative Property

## Definition

An operator or function op is associative if the following holds:

$$(a \text{ op } b) \text{ op } c == a \text{ op } (b \text{ op } c)$$

## Parallel

The importance of this to parallel evaluation can be seen if we expand this to four terms:

$$a \text{ op } b \text{ op } c \text{ op } d == (a \text{ op } b) \text{ op } (c \text{ op } d)$$

So we can evaluate <wrap box>(a op b)</note> in parallel with <wrap box>(c op d)</note>, and then invoke op on the results.

## Linear Algebra

Associativity of function composition: $h \circ (g \circ f) = (h \circ g) \circ f$

 $(h \circ (g \circ f))(x)$ = $h(g \circ f)(x)$ $h(g(f(x)))$ $(h \circ g)(f(x))$ $((h \circ g) \circ f)(x)$

## Example

Examples of associative operations include:

$$(x + y) + z = x + (y + z)$$