# Fully-characteristic congruence

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