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Difference between revisions of "Fractional congruence"

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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
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$#C+1 = 9 : ~/encyclopedia/old_files/data/F041/F.0401210 Fractional congruence
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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" />.
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The congruence  $  \eta / \theta $
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of a quotient system  $  \mathbf A / \theta $
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defined by the formula
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$$
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[ x ] _  \theta  ( \eta / \theta ) [ y ] _  \theta  \iff  x \eta y ,
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$$
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where  $  \eta $
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is some congruence of the [[Algebraic system|algebraic system]] $  \mathbf A $
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containing the given congruence $  \theta $
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and $  [ a] _  \theta  = \{ {x \in \mathbf A } : {x \theta a } \} $.  
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The quotient system $  ( \mathbf A / \theta ) / ( \eta / \theta ) $
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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=46969
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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