general associativity (original) (raw)

If an associative binary operation of a set S is denoted by “⋅”, the associative law in S is usually expressed as

or leaving out the dots, (a⁢b)⁢c=a⁢(b⁢c). Thus the common value of both may be denoted as a⁢b⁢c. With four elements of S we can , using only the associativity, as follows:

So we may denote the common value of those five expressions as a⁢b⁢c⁢d.

Theorem.

The expression formed of elements a1, a2, …, an of S . The common value is denoted by a1⁢a2⁢…⁢an.

Note. The n elements can be joined, without changing their , in (2⁢n-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∈ℤ+. The expression a⁢a⁢…⁢a with n equal “factors” a may be denoted by an and called a power of a. If the associative operation is denoted “additively”, then the “sum” a+a+⋯+a of n equal elements a is denoted by n⁢a 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:

In notation: