Congruence (in algebra)  From Encyclopedia of Mathematics Jump
to: navigation, search  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.