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Difference between revisions of "Power associativity"

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(Start article: Power associativity)
 
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A [[binary operation]] $\star$ on a set $X$ is ''power associative'' if it satisfies the condition
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A [[binary operation]] $\star$ on a set $X$ is ''power associative'' if each element $x$ generates an associative [[magma]]: that is, exponentiation $x \mapsto x^n$ is well-defined for positive integers $n$, and $x^{m+n} = x^m \star x^n$.  The set of powers of $x$ thus forms a [[semi-group]].
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$$
 
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x \star ( x \star x) = (x \star x) \star x
 
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$$
 
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for all $x \in X$.
 
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For such operations, exponentiation $x \mapsto x^n$ is well-defined for positive integers $n$, and $x^{m+n} = x^m \star x^n$.  The set of powers of $x$ forms a [[semi-group]].
 
  
 
See also: [[Algebra with associative powers]].
 
See also: [[Algebra with associative powers]].

Latest revision as of 10:28, 1 January 2016

A binary operation $\star$ on a set $X$ is power associative if each element $x$ generates an associative magma: that is, exponentiation $x \mapsto x^n$ is well-defined for positive integers $n$, and $x^{m+n} = x^m \star x^n$. The set of powers of $x$ thus forms a semi-group.

See also: Algebra with associative powers.

References

  • Bruck, Richard Hubert A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 20 Springer (1958) Zbl 0081.01704
How to Cite This Entry:
Power associativity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power_associativity&oldid=37214
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