View source for Fully-characteristic congruence

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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 $-
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====
<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====
Fully-characteristic congruences are also called fully-invariant congruences.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR></table>