# Congruence on an algebra

## 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 $A$ is an algebra with operators $\omega_j: A^{n_j} \to A$ (i.e., each operator $\omega_j$ has $n_j$ inputs and one output). A congruence on $A$ is an equivalence relation $\sim$ on $A$ such that, for all $j$:

$a_i \sim b_i, 1 \le i \le n_j \implies \omega_j(a_1,a_2,\dots,a_{n_j}) \sim \omega_j(b_1,b_2,\dots,b_{n_j})$

To every congruence, there is associated a natural quotient map, to the algebra $A/\sim$ of equivalence classes under $\sim$. If $A$ belongs to any variety, so does $A/\sim$, 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 $a \sim b \implies a^{-1} \sim b^{-1}$ and $a \sim b, c \sim d \implies ac \sim bd$. 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.