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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 $i=1,\ldots,n$.


A semi-group is a set equipped with an associative binary operation.

Weaker properties related to associativity include power associativity, the alternative identity and the flexible identity.


[a1] R.H. Bruck, "A survey of binary systems" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 20 Springer (1958) Zbl 0081.01704
How to Cite This Entry:
Associativity. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. IvanovaD.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article