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A congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f0419101.png" /> of an [[Algebraic system|algebraic system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f0419102.png" /> which is invariant under any endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f0419103.png" /> of this system, i.e. it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f0419104.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f0419105.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f0419106.png" />). The fully-characteristic congruences of an algebraic system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f0419107.png" /> form under inclusion a complete sublattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f0419108.png" /> of the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f0419109.png" /> of all congruences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191011.png" /> is a variety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191012.png" />-systems and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191013.png" /> is a [[Free algebra|free algebra]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191014.png" /> on a countably infinite set of generators, the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191015.png" /> of fully-characteristic congruences of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191016.png" /> is dually isomorphic to the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191017.png" /> of all subvarieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191018.png" />. Any congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191019.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191020.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191021.png" /> with a finite number of generators, of finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191022.png" /> (i.e. with a finite number of congruence classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191024.png" />), contains a fully-characteristic congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191026.png" /> which also has finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041910/f04191027.png" />.
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A congruence  $  \theta $
 +
of an [[Algebraic system|algebraic system]] $  A = \langle  A, \Omega \rangle $
 +
which is invariant under any endomorphism $  \sigma $
 +
of this system, i.e. it follows from $  x \theta y $
 +
that $  \sigma ( x) \theta \sigma ( y) $(
 +
$  x, y \in A $).  
 +
The fully-characteristic congruences of an algebraic system $  A $
 +
form under inclusion a complete sublattice $  C _ {v} ( A) $
 +
of the lattice $  C( A) $
 +
of all congruences of $  A $.  
 +
If $  \mathfrak M $
 +
is a variety of $  \Omega $-
 +
systems and if $  F $
 +
is a [[Free algebra|free algebra]] in $  \mathfrak M $
 +
on a countably infinite set of generators, the lattice $  C _ {v} ( F) $
 +
of fully-characteristic congruences of the system $  F $
 +
is dually isomorphic to the lattice $  L _ {v} ( \mathfrak M ) $
 +
of all subvarieties of $  \mathfrak M $.  
 +
Any congruence $  \kappa $
 +
of an $  \Omega $-
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algebra $  A $
 +
with a finite number of generators, of finite index in $  A $(
 +
i.e. with a finite number of congruence classes $  a/ \kappa $,  
 +
$  a \in A $),  
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contains a fully-characteristic congruence $  \theta $
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of $  A $
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which also has finite index in $  A $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:40, 5 June 2020


A congruence $ \theta $ of an algebraic system $ A = \langle A, \Omega \rangle $ which is invariant under any endomorphism $ \sigma $ of this system, i.e. it follows from $ x \theta y $ that $ \sigma ( x) \theta \sigma ( y) $( $ x, y \in A $). The fully-characteristic congruences of an algebraic system $ A $ form under inclusion a complete sublattice $ C _ {v} ( A) $ of the lattice $ C( A) $ of all congruences of $ A $. If $ \mathfrak M $ is a variety of $ \Omega $- systems and if $ F $ is a free algebra in $ \mathfrak M $ on a countably infinite set of generators, the lattice $ C _ {v} ( F) $ of fully-characteristic congruences of the system $ F $ is dually isomorphic to the lattice $ L _ {v} ( \mathfrak M ) $ of all subvarieties of $ \mathfrak M $. Any congruence $ \kappa $ of an $ \Omega $- algebra $ A $ with a finite number of generators, of finite index in $ A $( i.e. with a finite number of congruence classes $ a/ \kappa $, $ a \in A $), contains a fully-characteristic congruence $ \theta $ of $ A $ which also has finite index in $ A $.

References

[1] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)

Comments

Fully-characteristic congruences are also called fully-invariant congruences.

References

[a1] P.M. Cohn, "Universal algebra" , Reidel (1981)
How to Cite This Entry:
Fully-characteristic congruence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fully-characteristic_congruence&oldid=47006
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article