# Associativity

In algebra, **associativity** is a property of binary operations. If is a binary operation then the associative property is the condition that

for all *x*, *y* and *z*.

Examples of associative operations are addition and multiplication of integers, rational numbers, real and complex numbers. In this context associativity is often referred to as the *associative law*. Function composition is associative.

An important example of an algebraic structure in which the multiplication is not associative is the octonions.

## Related properties

An operation is **left alternative** if

for all *x* and *y*: it is **right alternative** if

An operation is **power-associative** if

for all *x*. In such cases the expression is well-defined for all positive integers *n*.

## Operator associativity

When an operation is not associative a convention is required to disambiguate an expression such as
*x* * *y* * *z*, and this convention may be described as the associativity of the operator "*". *Left-to-right* associativity means that the expression is to be interpreted as (*x* * *y*) * *z* (which is the normal arithmetical convention for subtraction and division) and *right-to-left* associativity means *x* * (*y* * *z*) (which is the normal arithmetical convention for exponentiation). Such conventions may be important in computer programming languages where a mathematically associative operator may have a non-associative numerical implementation.

## References

- Richard D. Schafer (1995).
*An introduction to Non-associative algebras*. Dover Publications, 1-8. ISBN 0-486-68813-5.