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