Congruence on an algebra

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.

On a group
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.