Fully-characteristic congruence
A congruence 
of an algebraic system 
which is invariant under any endomorphism 
of this system, i.e. it follows from 
that 
(
). The fully-characteristic congruences of an algebraic system 
form under inclusion a complete sublattice 
of the lattice 
of all congruences of 
. If 
is a variety of 
-systems and if 
is a free algebra in 
on a countably infinite set of generators, the lattice 
of fully-characteristic congruences of the system 
is dually isomorphic to the lattice 
of all subvarieties of 
. Any congruence 
of an 
-algebra 
with a finite number of generators, of finite index in 
(i.e. with a finite number of congruence classes 
, 
), contains a fully-characteristic congruence 
of 
which also has finite index in 
.
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) |
Fully-characteristic congruence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fully-characteristic_congruence&oldid=47006