# 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 $a_{1}a_{2}\ldots a_{n}$.

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

The theorem is proved by induction on $n$.  The cases  $n=3$  and  $n=4$  have been stated above.

Let  $n\in\mathbb{Z}_{+}$.  The expression $aa\ldots a$ 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”  $a\!+\!a\!+\cdots+\!a$  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