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 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 expressions as $abcd$ .

Note. The $n$ elements can be joined, without changing their order, 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 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 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 additive 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 force.