Associative law
The associative law ( Latin associare "unite, connect, link, network"), in German linkage law or connection law , is a rule from mathematics . A ( twodigit ) link is associative if the order of execution is not important. In other words: The bracketing of several associative links is arbitrary. That is why it can be clearly called the “law of brackets”.
In addition to the associative law, the commutative law and the distributive law are of elementary importance in algebra .
definition
A binary connection on a set is called associative if the associative law applies to all
applies. The brackets can then be omitted. This also applies to more than three operands .
Examples and counterexamples
As links on the real numbers , addition and multiplication are associative. For example
and
. 
Real subtraction and division , however, are not associative because it is
and
. 
The potency is not associative either
applies. In the case of (divergent) infinite sums , the brackets may be important. Addition loses associativity for:
but
classification
The associative law belongs to the group axioms , but is already required for the weaker structure of a semigroup .
Laterality
In the case of nonassociative links in particular, there are conventions of side associativity.
A binary link is considered to be leftassociative if
is to be understood.
 The nonassociative operations subtraction and division are commonly understood to be leftassociative:
( Subtraction )  
( Division ) 
 Application of functions in the process of currying .
A binary connection is called rightassociative if the following applies:
Example of a rightassociative operation:
 Exponentiation of real numbers in exponent notation:
But associative operations can also have sidedness if they can be iterated to infinity.
 The decimal notation right of the decimal is a leftassociative linking of the decimal number, because the evaluation (sschleife) is not right with the ellipsis can begin, but must start on the left.
 The adic notation contains a rightassociative chaining operation with the juxtaposition , because the evaluation must begin on the right.
Weaker forms of associativity
The following weakening of the associative law is mentioned / defined elsewhere:

Potency Associativity :

Alternative :

Left alternative :

Legal alternativity :

Left alternative :

Flexibility law :

Moufang identities :
 Bol identities:

left Bol identity:

right Bol identity:

left Bol identity:

Jordan Identity :
See also
literature
 Otto Forster: Analysis 1: differential and integral calculus of a variable . ViewegVerlag, Munich 2008, ISBN 9783834803955 .
Individual evidence
 ^ Bronstein: Pocket book of mathematics . Chapter: 2.4.1.1, ISBN 9783808556733 , pp. 115120
 ^ George Mark Bergman: Order of arithmetic operations
 ^ The Order of Operations. Education Place
 ↑ Gerrit Bol: Web and Groups In: Mathematische Annalen , 114 (1), 1937, pp. 414431.