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Difference between revisions of "Associativity"

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''law of associativity''
 
''law of associativity''
  
A property of an [[Algebraic operation|algebraic operation]]. For the addition and multiplication of numbers associativity is expressed by the following identities:
+
A property of an [[Algebraic operation|algebraic operation]]. For the addition and multiplication of numbers, associativity is expressed by the following identities:
 +
$$
 +
a+(b+c) = (a+b) + c\ \ \text{and}\ \ a(bc) = (ab)c\ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013520/a0135201.png" /></td> </tr></table>
+
A general [[binary operation]] $\star$ is associative (or, which is the same thing, satisfies the law of associativity) if the identity
 +
$$
 +
a \star (b \star c) = (a \star b) \star c
 +
$$
 +
is valid in the given algebraic system. In a similar manner, associativity of an $n$-ary operation $\omega$ is defined by the identities
 +
$$
 +
(x_1 x_2 \ldots x_n)\omega x_{n+1} \ldots x_{2n-1} \omega = x_1 \ldots x_i (x_{i+1} \ldots x_{i+n})\omega x_{i+n+1} \ldots x_{i+2n-1} \omega
 +
$$
 +
for all $i=1,\ldots,n$.
  
A binary algebraic operation * is associative (or, which is the same thing, satisfies the law of associativity) if the identity
+
{{TEX|done}}
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013520/a0135202.png" /></td> </tr></table>
 
 
 
is valid in the given algebraic system. In a similar manner, associativity of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013520/a0135203.png" />-ary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013520/a0135204.png" /> is defined by the identities
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013520/a0135205.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013520/a0135206.png" /></td> </tr></table>
 
 
 
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013520/a0135207.png" />.
 

Revision as of 20:46, 7 January 2016

law of associativity

A property of an algebraic operation. For the addition and multiplication of numbers, associativity is expressed by the following identities: $$ a+(b+c) = (a+b) + c\ \ \text{and}\ \ a(bc) = (ab)c\ . $$

A general binary operation $\star$ is associative (or, which is the same thing, satisfies the law of associativity) if the identity $$ a \star (b \star c) = (a \star b) \star c $$ is valid in the given algebraic system. In a similar manner, associativity of an $n$-ary operation $\omega$ is defined by the identities $$ (x_1 x_2 \ldots x_n)\omega x_{n+1} \ldots x_{2n-1} \omega = x_1 \ldots x_i (x_{i+1} \ldots x_{i+n})\omega x_{i+n+1} \ldots x_{i+2n-1} \omega $$ for all $i=1,\ldots,n$.

How to Cite This Entry:
Associativity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Associativity&oldid=17122
This article was adapted from an original article by O.A. IvanovaD.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article