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Verbal congruence

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A congruence on an algebra which is expressible as the intersection of all congruences on whose factor algebras belong to some fixed variety of -algebras. A congruence on an arbitrary algebraic system is said to be verbal if there exists a variety of -systems for which the canonical mapping is universal amongst the morphisms from to algebras in . A verbal congruence is a fully-characteristic congruence. If is a free -system in some variety , then, conversely, any fully-characteristic congruence in is a verbal congruence with respect to the variety generated by the factor system .

References

[1] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)


Comments

Cf. also Universal property; Congruence (in algebra).

How to Cite This Entry:
Verbal congruence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Verbal_congruence&oldid=35793
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article