associative

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

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

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 $\nu (\nu (a,b),c)=\nu (a^2+b,c)=a^4+2b{a}^2+b^2+c$. But $\nu (a,\nu (b,c))=\nu (a,b^2+c)=a+b^4+2c{b}^2+c^2$, 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 $\varphi$ on a set $A$ is said to be associative if for any elements $a_1,\mathrm{\dots },a_{2n-1}\in A$, we have

$$\varphi (\varphi (a_1,\mathrm{\dots },a_n),a_{n+1}\mathrm{\dots },a_{2n-1})=\mathrm{\cdots }=\varphi (a_1,\mathrm{\dots },a_{n-1},\varphi (a_n,\mathrm{\dots },a_{2n-1})).$$

In other words, for any $i=1,\mathrm{\dots },n$, if we set $b_i:=\varphi (a_1,\mathrm{\dots },\varphi (a_i,\mathrm{\dots },a_{i+n-1}),\mathrm{\dots },a_{2n-1})$, then $\varphi$ is associative iff $b_i=b_1$ for all $i=1,\mathrm{\dots },n$. Therefore, for instance, a ternary operation $f$ on $A$ is associative if $f(f(a,b,c),d,e)=f(a,f(b,c,d),e)=f(a,b,f(c,d,e))$.