# 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)