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
.