# Associativity

law of associativity

A property of an algebraic operation. For the addition and multiplication of numbers, associativity is expressed by the following identities: $$a+(b+c) = (a+b) + c\ \ \text{and}\ \ a(bc) = (ab)c\ .$$

A general binary operation $\star$ is associative (or, which is the same thing, satisfies the law of associativity) if the identity $$a \star (b \star c) = (a \star b) \star c$$ is valid in the given algebraic system. In a similar manner, associativity of an $n$-ary operation $\omega$ is defined by the identities $$(x_1 x_2 \ldots x_n)\omega x_{n+1} \ldots x_{2n-1} \omega = x_1 \ldots x_i (x_{i+1} \ldots x_{i+n})\omega x_{i+n+1} \ldots x_{i+2n-1} \omega$$ for all $i=1,\ldots,n$.