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%%% Primary Title: general associativity
%%% Primary Category Code: 20-00
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%%% Owner: pahio
%%% Author(s): pahio
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\begin{document}

 If an \htmladdnormallink{associative}{http://planetmath.org/encyclopedia/Associative.html} \htmladdnormallink{binary operation}{http://planetmath.org/encyclopedia/InternalComposition.html} 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 sides may be denoted as $abc$.\, With four elements of $S$ we can calculate, 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 \htmladdnormallink{expressions}{http://planetmath.org/encyclopedia/Expression.html} as $abcd$.

\begin{thmplain}
\, The expression formed of elements $a_1$, $a_2$, \ldots, $a_n$ of $S$ represents always the same element of $S$ independently on how one has joined them together with the associative operation and parentheses, if only the order of the elements is every time the same.\, The common value is denoted by $a_1a_2\ldots a_n$.
\end{thmplain}

\textbf{Note.}\, The $n$ elements can be joined, without changing their order, in $\frac{(2n-2)!}{n!(n-1)!}$ ways (see the \htmladdnormallink{Catalan numbers}{http://planetmath.org/encyclopedia/CatalanNumbers.html}).

The theorem is proved by \htmladdnormallink{induction}{http://planetmath.org/encyclopedia/PartialWellOrder.html} on $n$.\, The cases\, $n = 3$\, and\, $n = 4$\, have been stated right 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 {\em power} of $a$.\, If the associative \htmladdnormallink{operation}{http://planetmath.org/encyclopedia/Operation.html} is denoted ``additively'', then the ``sum''\,
$a\!+\!a\!+\cdots+\!a$\, of $n$ equal elements $a$ is denoted by $na$ and called a {\em multiple} of $a$; hence in every \htmladdnormallink{ring}{http://planetmath.org/encyclopedia/UnitalRing.html} one may consider powers and multiples. According to whether $n$ is an \htmladdnormallink{even}{http://planetmath.org/encyclopedia/OddInteger.html} or an \htmladdnormallink{odd number}{http://planetmath.org/encyclopedia/OddInteger.html}, one may speak of {\em even powers}, {\em odd powers}, {\em even multiples}, {\em 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 additive notation:
$$ma\!+\!na = (m\!+\!n)a,$$
$$n(ma) = (mn)a$$

\textbf{Note.}\, If the set $S$ together with its operation is a \htmladdnormallink{group}{http://planetmath.org/encyclopedia/NontrivialElement2.html}, then the notion of multiple $na$ resp. power $a^n$ can be extended for \htmladdnormallink{negative}{http://planetmath.org/encyclopedia/Negative.html} \htmladdnormallink{integer}{http://planetmath.org/encyclopedia/RationalInteger.html} and zero values of $n$ by means of the \htmladdnormallink{inverse}{http://planetmath.org/encyclopedia/NontrivialElement2.html} and \htmladdnormallink{identity elements}{http://planetmath.org/encyclopedia/IdentityElement.html}.\, The above laws remain in force.

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