Pre-additive categories

Previously, we talked about groups with additonal structure. A more general concept is that of a category with additional structure, the most important example of which is a pre-additive category. Pre-additive categories are categories whose hom classes are commutative groups, such that the group distributes over composition. A (not necessarily commutative) ring is a pre-additive category with a single object.

Non-commutative rings of actions:
Let $C$ be a concrete category and $X \in Ob(C)$ then $End(X)$ is a monoid action on the underlying set of $X$. A wide variety of different monoid actions emerge from concrete categories like posets, graphs, etc. We can also explore the endomorphism monoid of a commutative group. In this case, pre-additive categories add additional perspective to the theory of commutative groups.

By taking the category of commutative groups to be an abelian category, we get that $End(G)$ for any commutative group is an endomorphism ring. Therefore the actions which previously only took on the structure of a monoid, now have the structure of a ring. This is the basic context in which non-commutative rings emerge and become indispensible. We naturally consider rings to be commutative starting with the ring of integers $\mathbb{Z}$, but here we see how some non-commutative rings emerge even in problems of commutative algebra.

Let $R$ be a ring, then the category of $R$-modules is an abelian category. In this case, for any $R$-module $M$ we have that $End(M)$ is the matrix ring of the module $M$. A more familiar case might be the matrix ring of a vector space over a field. These matrix rings are amongst the most important non-commutative rings, and they emerge as rings of actions on a set. Therefore, non-commutative rings are an indispensible part of the modern algebraic theory of actions.

Concrete rings:
A concrete pre-additive category is a pre-additive category $C$ with a faithful set-valued functor $F : C \to Sets$. This makes it so that all the elements of the ring are functions acting on a set. In this case, we have a multiplicative monoid action on the underlying set of the concrete ring. We see that the matrix ring $End(V)$ is a concrete ring whose elements are functions acting on the underlying set of vectors. Concrete rings proide a natural categorical description of non-commutative rings of actions.

Links:
Preadditive and additive categories

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