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# 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 $abcd$.

###### Theorem.

The expression formed of elements $a_{1}$, $a_{2}$, …, $a_{n}$ 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 $a_{1}a_{2}\ldots a_{n}$.

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

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:

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

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

## Mathematics Subject Classification

20-00*no label found*

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