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Fully-characteristic congruence

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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)
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