general associativity

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


or leaving out the dots,  (ab)c=a(bc).  Thus the common value of both may be denoted as abc.  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 abcd.


The expression formed of elements a1, a2, …, an of S .  The common value is denoted by a1a2an.

Note.  The n elements can be joined, without changing their , in (2n-2)!n!(n-1)! ways (see the Catalan numbersMathworldPlanetmath).

The theorem is proved by inductionMathworldPlanetmath on n.  The cases  n=3  and  n=4  have been stated above.

Let  n+.  The expression aaa with n equal “factors” a may be denoted by an and called a power of a.  If the associative operationMathworldPlanetmath is denoted “additively”, then the “sum”  a+a++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:


In notation:


Note.  If the set S together with its operation is a group, then the notion of multiple na resp. power an can be extended for negative integer and zero values of n by means of the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and identity elementsMathworldPlanetmath.  The above laws remain in .

Title general associativity
Canonical name GeneralAssociativity
Date of creation 2013-03-22 14:35:50
Last modified on 2013-03-22 14:35:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 21
Author pahio (2872)
Entry type Theorem
Classification msc 20-00
Related topic SemigroupPlanetmathPlanetmath
Related topic EveryRingIsAnIntegerAlgebra
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Related topic Characteri
Defines power
Defines multiple
Defines even power
Defines odd power
Defines even multiple
Defines odd multiple