Congruence (in algebra)
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) |
Congruence (in algebra). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruence_(in_algebra)&oldid=39720