# Congruence on an algebra

## Contents

## Definition

### Symbol-free definition

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.

## Examples

### On a group

`Further information: congruence on a group, normal subgroup equals kernel of homomorphism, first isomorphism theorem`

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.