Associativity

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

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

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

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

An operation is power-associative if

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

for all x. In such cases the expression $x^{n}$ 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.