or leaving out the dots, . Thus the common value of both may be denoted as . With four elements of we can , using only the associativity, as follows:
So we may denote the common value of those five expressions as .
The expression formed of elements , , …, of . The common value is denoted by .
Note. The elements can be joined, without changing their , in ways (see the Catalan numbers).
Let . The expression with equal “factors” may be denoted by and called a power of . If the associative operation is denoted “additively”, then the “sum” of equal elements is denoted by and called a multiple of ; hence in every ring one may consider powers and multiples. According to whether 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:
Note. If the set together with its operation is a group, then the notion of multiple resp. power can be extended for negative integer and zero values of by means of the inverse and identity elements. The above laws remain in .
|Date of creation||2013-03-22 14:35:50|
|Last modified on||2013-03-22 14:35:50|
|Last modified by||pahio (2872)|