Let $(S,\phi)$ be a set with binary operation $\phi$ . $\phi$ is said to be associative over $S$ if

$$ \phi(a,\phi(b,c)) = \phi(\phi(a,b),c) $$

for all $a,b,c \in S$ .

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

We can construct an operation which is not associative. Let $S$ be the integers. and define $\nu(a,b)=a^2+b$ . Then $\nu(\nu(a,b),c)=\nu(a^2+b,c)=a^4+2ba^2+b^2+c$ . But $\nu(a,\nu(b,c))=\nu(a,b^2+c)=a+b^4+2cb^2+c^2$ , hence $\nu(\nu(a,b),c) \ne \nu(a,\nu(b,c))$ .

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

Example.

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 $n\ge 2$ . An $n$ -ary operation $\phi$ on a set $A$ is said to be associative if for any elements $a_1,\ldots, a_{2n-1} \in A$ , we have $$\phi(\phi(a_1,\ldots, a_n), a_{n+1} \ldots, a_{2n-1}) = \cdots = \phi(a_1, \ldots, a_{n-1}, \phi(a_n,\ldots, a_{2n-1})).$$ In other words, for any $i=1,\ldots, n$ , if we set $b_i:=\phi (a_1,\ldots, \phi(a_i,\ldots, a_{i+n-1}),\ldots, a_{2n-1})$ , then $\phi$ is associative iff $b_i=b_1$ for all $i=1,\ldots, n$ . Therefore, for instance, a ternary operation $f$ on $A$ is associative if $f(f(a,b,c),d,e)=f(a,f(b,c,d),e)=f(a,b,f(c,d,e))$ .