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