Congruences of structured groups

If we have a group $G$ then all congruences of $G$ can be determined by its sublattice of normal subgroups. This is determined by the action preorders of normal subgroups, which are symmetric preorders (equivalence relations), and congruences of $G$ by normality.

Suppose that $(S,+,...)$ is a group $(S,+)$ with some additional structure $(...)$ on it. Then we can use the same procedure to get group congruence of $+$ from normal subgroups but these congruences need not be congruences of $S$. In order to get full congruences of the structured group, we need a special subset of congruence-forming normal subgrops. These are the subset of normal subgroups that produce congruences by their cosets.

Definition. let $(S,+,...)$ be a group with additional structure. Then a congruence-producing normal subgroup is a normal subgroup of $+$ whose cosets are a congruence of the entire structure $S$.

We can use this to formalize the congruencization mapping \[f: (K \subseteq Sub(S)) \to Con(S) \] by setting $K$ equal to the family of all congruence-producing normal subgroups of the structured group. Then this converts the normal subgroup into its congruence by taking cosets. In the opposite direction we have a monomorphism: \[ g : Con(S) \to Sub(S) \] This monomorphism is an order-embedding from the lattice of congruences into the lattice of subalgebras. In the simplest case of a group, the lattice of congruences of the group can be embedded in the sublattice of normal subgroups. The function of the monomorphism $g$ is determined by taking the kernel subalgebra of the congruence, which is the unique additive identity element in the quotient algebra.

In commutative algebra we have that ideals form congruences. On the other hand, in non-commutative algebra we have that two-sided ideals (distinguished from right and left ideals by non-commutativity) form congruences. Likewise, submodules determine congruences.

Groups, commutative rings, non-commutative rings, modules, vector spaces, etc are all examples of structured groups. Categories of structured groups are distinguished by the existence of kernel representations of congruences. Most abelian categories in homological algebra for example emerge from structured groups like R-modules. Even in the case of sheaves of commutative groups, the abelian category is constructed from some underlying group structure.

The need to form a separate theory of congruences emerges when considering semigroups, semirings, and other semi-structures that don't have any group operations assigned to them. Classical abstract algebra avoids this by assigning a group to everything, but a particularly interesting new direction is the general theory of structured categories. General structured categories require a separate theory of congruences. Monoid actions are another application of congruences, which can play a central role in the algebraic theory of computation.

Previously:
Subalgebra related congruences