# 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 $f(f(a,b,c),d,e)=f(a,f(b,c,d),e)=f(a,b,f(c,d,e))$.

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 | Commutative^{} |

Related topic | Semigroup^{} |

Related topic | Group |

Defines | non-associative |