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