Congruence on an algebra
A congruence on an algebra is an equivalence relation that is preserved by all the operations of the algebra.
Definition with symbols
Suppose is an algebra with operators (i.e., each operator has inputs and one output). A congruence on is an equivalence relation on such that, for all :
To every congruence, there is associated a natural quotient map, to the algebra of equivalence classes under . If belongs to any variety, so does , so we can study the notion of congruence restricted to a particular variety of algebras.
On a group
A congruence on a group is an equivalence relation such that and . For any congruence, the corresponding quotient gives a quotient map in the usual sense, and the kernel of the quotient map, which is the set of elements congruent to the identity, is a normal subgroup.
Conversely, given any normal subgroup, there is a unique congruence having that as kernel. (The uniqueness is another formulation of the first isomorphism theorem). The universal algebraic statement for this is that the variety of groups is ideal-determined.
On a monoid
A congruence on a monoid is an equivalence relation that respects the monoid multiplication. We can also define a quotient map with respect to the congruence.
However, the set of elements congruent to the identity does not determine the congruence completely.