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单词 GeneralAssociativity
释义

general associativity


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

(ab)c=a(bc),

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:

(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 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:

aman=am+n
(am)n=amn

In 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 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 .

Titlegeneral associativity
Canonical nameGeneralAssociativity
Date of creation2013-03-22 14:35:50
Last modified on2013-03-22 14:35:50
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id21
Authorpahio (2872)
Entry typeTheorem
Classificationmsc 20-00
Related topicSemigroupPlanetmathPlanetmath
Related topicEveryRingIsAnIntegerAlgebra
Related topicInverseFormingInProportionToGroupOperation
Related topicCosineAtMultiplesOfStraightAngle
Related topicInfixNotation
Related topicOperationsOnRelations
Related topicDifference2
Related topicFactorsWithMinusSign
Related topicIdealOfElementsWithFiniteOrder
Related topicGeneralCommutativity
Related topicCharacteri
Definespower
Definesmultiple
Defineseven power
Definesodd power
Defineseven multiple
Definesodd multiple
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更新时间:2025/5/4 21:19:39