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Difference between revisions of "Principal translation"

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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074780/p0747801.png" /> of an [[Algebraic system|algebraic system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074780/p0747802.png" /> into itself, of the form
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''Elementary translation''
  
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<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/p/p074/p074780/p0747803.png" /></td> </tr></table>
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A mapping $\phi$ of an [[algebraic system]] $\mathbf{A} = (A,\Omega)$ into itself, of the form
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$$
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\phi : x \mapsto F(a_1,\ldots,a_{k-1},x,a_{k+1},\ldots,a_n)
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$$
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where $F$ is the symbol of a basic operation in $\Omega$ and $a_1,\ldots,a_n$ are fixed elements of the set $A$.
  
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074780/p0747804.png" /> is the symbol of a basic operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074780/p0747805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074780/p0747806.png" /> are fixed elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074780/p0747807.png" />.
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The terminology "elementary translation" is also used: as are "algebraic function" (of one variable) or "polynomial".
  
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====References====
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* Cohn, Paul M. Universal algebra. Rev. ed. D. Reidel (1981) {{ISBN|90-277-1213-1}} {{ZBL|0461.08001}}
  
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{{TEX|done}}
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====Comments====
 
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The term  "principal translation"  is not used in the Western literature. A function as above would normally be called an algebraic function (of one variable) or a polynomial.
 

Latest revision as of 16:59, 23 November 2023

Elementary translation

A mapping $\phi$ of an algebraic system $\mathbf{A} = (A,\Omega)$ into itself, of the form $$ \phi : x \mapsto F(a_1,\ldots,a_{k-1},x,a_{k+1},\ldots,a_n) $$ where $F$ is the symbol of a basic operation in $\Omega$ and $a_1,\ldots,a_n$ are fixed elements of the set $A$.

The terminology "elementary translation" is also used: as are "algebraic function" (of one variable) or "polynomial".

References

  • Cohn, Paul M. Universal algebra. Rev. ed. D. Reidel (1981) ISBN 90-277-1213-1 Zbl 0461.08001
How to Cite This Entry:
Principal translation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_translation&oldid=13815
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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