# general associativity

If an associative binary operation^{} of a set $S$ is denoted by “$\cdot $”, the associative law in $S$ is usually expressed as

$$(a\cdot b)\cdot c=a\cdot (b\cdot c),$$ |

or leaving out the dots, $(ab)c=a(bc)$. Thus the common value of both may be denoted as $abc$. With four elements of $S$ we can , using only the associativity, as follows:

$$(ab)(cd)=a(b(cd))=a((bc)d)=(a(bc))d=((ab)c)d$$ |

So we may denote the common value of those five expressions as $abcd$.

###### Theorem.

The expression formed of elements ${a}_{1}$, ${a}_{2}$, …, ${a}_{n}$ of $S$ . The common value is denoted by ${a}_{1}{a}_{2}\mathrm{\dots}{a}_{n}$.

Note. The $n$ elements can be joined, without changing their , in $\frac{(2n-2)!}{n!(n-1)!}$ ways (see the Catalan numbers^{}).

Let $n\in {\mathbb{Z}}_{+}$. The expression $aa\mathrm{\dots}a$ with $n$ equal “factors” $a$ may be denoted by ${a}^{n}$ and called a power of $a$. If the associative operation^{} is denoted “additively”, then the “sum”
$a+a+\mathrm{\cdots}+a$ of $n$ equal elements $a$ is denoted by $na$ and called a multiple of $a$; hence in every ring one may consider powers and multiples. According to whether $n$ is an even or an odd number, one may speak of even powers, odd powers, even multiples, odd multiples.

The following two laws can be proved by induction:

$${a}^{m}\cdot {a}^{n}={a}^{m+n}$$ |

$${({a}^{m})}^{n}={a}^{mn}$$ |

In notation:

$$ma+na=(m+n)a,$$ |

$$n(ma)=(mn)a$$ |

Note. If the set $S$ together with its operation is a group, then the notion of multiple $na$ resp. power ${a}^{n}$ can be extended for negative integer and zero values of $n$ by means of the inverse^{} and identity elements^{}. The above laws remain in .

Title | general associativity |

Canonical name | GeneralAssociativity |

Date of creation | 2013-03-22 14:35:50 |

Last modified on | 2013-03-22 14:35:50 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 21 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 20-00 |

Related topic | Semigroup^{} |

Related topic | EveryRingIsAnIntegerAlgebra |

Related topic | InverseFormingInProportionToGroupOperation |

Related topic | CosineAtMultiplesOfStraightAngle |

Related topic | InfixNotation |

Related topic | OperationsOnRelations |

Related topic | Difference2 |

Related topic | FactorsWithMinusSign |

Related topic | IdealOfElementsWithFiniteOrder |

Related topic | GeneralCommutativity |

Related topic | Characteri |

Defines | power |

Defines | multiple |

Defines | even power |

Defines | odd power |

Defines | even multiple |

Defines | odd multiple |