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Congruence (in algebra)

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An equivalence relation on a universal algebra commuting with all operations in , that is, an equivalence relation such that whenever , where , , and is an -ary operation. Congruences in algebraic systems are defined in a similar way. Thus, the equivalence classes modulo a congruence form a universal algebra (algebraic system) of the same type as , called the quotient algebra (or quotient system) modulo . The natural mapping from onto (which takes an element to the -class containing it) is a surjective homomorphism. Conversely, every homomorphism defines a unique congruence, whose classes are the pre-images of the elements of .

The intersection of a family of congruences , , in the lattice of relations on a universal algebra (algebraic system) is a congruence. In general, a union of congruences in the lattice of relations is not a congruence. The product of two congruences and is a congruence if and only if and commute, i.e. if and only if .

References

[1] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)


Comments

References

[a1] P.M. Cohn, "Universal algebra" , Reidel (1981)
How to Cite This Entry:
Congruence (in algebra). V.S. Malakhovskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Congruence_(in_algebra)&oldid=12653
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098