Congruence on an algebra

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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.