Associativity

In algebra, associativity is a property of binary operations. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \star} is a binary operation then the associative property is the condition that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x \star y) \star z = x \star (y \star z) \,}

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

$(x\star x)\star y=x\star (x\star y)\,$

for all x and y: it is right alternative if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y \star x) \star x = y \star (x \star x) . \,}

An operation is power-associative if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x \star x) \star x = x \star (x \star x) \,}

for all x. In such cases the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^n} 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.