Let (S,ϕ) be a set with binary operationMathworldPlanetmath ϕ. ϕ is said to be associative over S if


for all a,b,cS.

Examples of associative operations are addition and multiplication over the integers (or reals), or addition or multiplication over n×n matrices.

We can construct an operation which is not associative. Let S be the integers. and define ν(a,b)=a2+b. Then ν(ν(a,b),c)=ν(a2+b,c)=a4+2ba2+b2+c. But ν(a,ν(b,c))=ν(a,b2+c)=a+b4+2cb2+c2, hence ν(ν(a,b),c)ν(a,ν(b,c)).

Note, however, that if we were to take S={0}, ν would be associative over S!. This illustrates the fact that the set the operation is taken with respect to is very important.


We show that the division operation over nonzero reals is non-associative. All we need is a counter-example: so let us compare 1/(1/2) and (1/1)/2. The first expression is equal to 2, the second to 1/2, hence division over the nonzero reals is not associative.

Remark. The property of being associative of a binary operation can be generalized to an arbitrary n-ary operation, where n2. An n-ary operation ϕ on a set A is said to be associative if for any elements a1,,a2n-1A, we have


In other words, for any i=1,,n, if we set bi:=ϕ(a1,,ϕ(ai,,ai+n-1),,a2n-1), then ϕ is associative iff bi=b1 for all i=1,,n. Therefore, for instance, a ternary operation f on A is associative if f(f(a,b,c),d,e)=f(a,f(b,c,d),e)=f(a,b,f(c,d,e)).

Title associative
Canonical name Associative
Date of creation 2013-03-22 12:22:48
Last modified on 2013-03-22 12:22:48
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 20-00
Synonym associativity
Related topic CommutativePlanetmathPlanetmathPlanetmath
Related topic SemigroupPlanetmathPlanetmath
Related topic Group
Defines non-associative