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).

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 $aa\mathrm{\dots }a$ 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