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# associative

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 $\nu(\nu(a,b),c)\neq\nu(a,\nu(b,c))$.

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 $n\geq 2$. 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 $f(f(a,b,c),d,e)=f(a,f(b,c,d),e)=f(a,b,f(c,d,e))$.

## Mathematics Subject Classification

20-00*no label found*

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