# (Function|Operator) - Associative Property

### Table of Contents

## Definition

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

<MATH> (a \text{ op } b) \text{ op } c == a \text{ op } (b \text{ op } c) </MATH>

## Articles Related

## Parallel

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

<MATH> a \text{ op } b \text{ op } c \text{ op } d == (a \text{ op } b) \text{ op } (c \text{ op } d) </MATH>

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: <math>h \circ (g \circ f) = (h \circ g) \circ f</math>

<math>(h \circ (g \circ f))(x)</math> = | <math>h(g \circ f)(x)</math>
<math>h(g(f(x)))</math> <math>(h \circ g)(f(x))</math> <math>((h \circ g) \circ f)(x)</math> |

## Example

Examples of associative operations include:

<MATH> (x + y) + z = x + (y + z) </MATH>