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 $j$:

$a_{i}\sim b_{i},1\leq i\leq n_{j}\implies \omega _{j}(a_{1},a_{2},\dots ,a_{n_{j}})\sim \omega _{j}(b_{1},b_{2},\dots ,b_{n_{j}})$

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.