Namespaces
Variants
Actions

Difference between revisions of "Fractional congruence"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
−
The congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041210/f0412101.png" /> of a quotient system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041210/f0412102.png" /> defined by the formula
+
<!--
 +
f0412101.png
 +
$#A+1 = 9 n = 0
 +
$#C+1 = 9 : ~/encyclopedia/old_files/data/F041/F.0401210 Fractional congruence
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
−
<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/f/f041/f041210/f0412103.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
−
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041210/f0412104.png" /> is some congruence of the [[Algebraic system|algebraic system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041210/f0412105.png" /> containing the given congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041210/f0412106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041210/f0412107.png" />. The quotient system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041210/f0412108.png" /> is isomorphic to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041210/f0412109.png" />.
+
The congruence  $  \eta / \theta $
 +
of a quotient system  $  \mathbf A / \theta $
 +
defined by the formula
 +
 
 +
$$
 +
[ x ] _  \theta  ( \eta / \theta ) [ y ] _  \theta  \iff  x \eta y ,
 +
$$
 +
 
 +
where  $  \eta $
 +
is some congruence of the [[Algebraic system|algebraic system]] $  \mathbf A $
 +
containing the given congruence $  \theta $
 +
and $  [ a] _  \theta  = \{ {x \in \mathbf A } : {x \theta a } \} $.  
 +
The quotient system $  ( \mathbf A / \theta ) / ( \eta / \theta ) $
 +
is isomorphic to the system $  \mathbf A / \eta $.

Latest revision as of 19:39, 5 June 2020


The congruence $ \eta / \theta $ of a quotient system $ \mathbf A / \theta $ defined by the formula

$$ [ x ] _ \theta ( \eta / \theta ) [ y ] _ \theta \iff x \eta y , $$

where $ \eta $ is some congruence of the algebraic system $ \mathbf A $ containing the given congruence $ \theta $ and $ [ a] _ \theta = \{ {x \in \mathbf A } : {x \theta a } \} $. The quotient system $ ( \mathbf A / \theta ) / ( \eta / \theta ) $ is isomorphic to the system $ \mathbf A / \eta $.

How to Cite This Entry:
Fractional congruence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractional_congruence&oldid=13683
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Fractional_congruence&oldid=46969"