Congruence lattices of quivers

Let $Q$ be a multi-directed graph, then $Q$ is associated to a lattice of congruences $Con(Q)$ which can be constructed using new and original algorithms of mine. A quiver can be seen as a presheaf over the following category: This category I call the double arrow category, because it has a repeated pair of arrows going from the first object to the second one. As a category, it has an underlying quiver $Q$. Its congruence lattice $Con(Q)$ looks like this: This defines the congruence lattice of a multi-directed graph, but the same could be applied to any directed graph without repeated edges. Consider the directed cycle graph $C_3$ on three elements: Then the congruence lattice of the directed cycle graph $Con(C_3)$ has this interesting structure: This different sort of congruence lattice for $Con(C_3)$ might look unexpected at first, but actually it makes perfect sense. It is formed by adjoining two different five elements congruence lattices of a three element set to one another. Recall that the congruence lattice $Con(S)$ of a set with three elements has the structure [1,3,1] and that it has five elements. The reason for this appearance is that the cycle $(0,1),(1,2),(2,0)$ has this unique property is that every vertex is in any two of its edges. Therefore, in order to form any congruence of $C_3$ you must first collapse all of its vertices.

We have shown that every directed multigraph is associated with a congruence lattice $Con(Q)$, but let us not forget that every object of a congruence lattice is associated with a quotient. In the case of $C_3$ all of the different congruences of the same size and height have the same quotient, and so all the different quotients of $C_3$ can be formed one after another. They are formed in five steps: starting with $C_3$, collapsing a pair of objects, then collapsing another pair of objects, then collapsing a pair of morphisms, then collpasing the last pair of morphisms.

We have now seen the congruence lattice $Con(C_3)$ of a directed cycle graph on three elements as well as all of its quotients. It had the unique structure of a weak order. It would be interesting to see if $C_4$ has the same structure of congruences: Then the congruence lattice of $Con(C_4)$ looks as displayed below. It does not have the property that it is two partition lattices adjoined to one another, because it doesn't have the property that any vertex is contained in any pair of edges. Nonetheless, you still have to collapse at least three objects before you can collapse one edge, so you can still visibly see the fifteen element partition lattice on four elements on the bottom connected to another one above it but now with some mixed congruences between them:

Another directed graph with an interesting congruence lattice is the strict total order $T_3$. It has a congruence lattice $Con(T_3)$ as displayed below. The reason for this interesting structure is that when you collapse two lower objects you then get repeated edges from the lower class to the upper one. These repeated edges can then be collapsed without equating more objects, and the same works in the other direction from above. So you get this interesting self dual structure in the form of a lattice. We have considered some interesting cases of directed graphs, but what if you have a repeated edges. In that case, we can see that repeated edges actually do have an effect on the appearance of the congruence lattice. A directed multigraph with repeated edges has as an atomic congruence a partition that equates two parallel edges rather then two vertices. So the atoms of such a congruence lattice don't necessarily generate a partition lattice, so they appear different. This is a non-thin quiver, so it has an atomic congruence that is not part of any partition lattice. It is the unique atom that is not a part of an element that covers three atoms. Consider two disconnected edges: Then their congruence lattice is displayed below. Initially, it just looks like a partition lattice on four elements, but there is a little bit more that meets the eye. There is one special congruence on the disconnected pair of edges: the one that equates both the minimal edges and that equates both the maximal edges. Then the two arrows can also be equated to get a parallel pair between two objects as a quotient. The disjoint arrows map is a function, but it is perhaps not a transformation because it is not closed. But we can make it one like this: Simply adding these two edges creates a much larger congruence lattice, which demonstrates how these congruence lattices tend to grow quite large for even small directed graphs: Topos theory generates congruence lattices for far more objects then classical lattice theory, which would only generated them for lattices or semilattices. We can now generate them for any poset. Consider the following poset: This poset is known as $[1,2]$ and its congruence lattice $Con([1,2])$ is of the following form: These congruence lattices are already getting quite larger. We can certainly go deeper, and create congruence lattices for even larger directed graphs and we can even run computations on them, but they will get too big for Graphviz to display nicely I think so we'll have to leave it at this. The congruence lattices generted so far should give you an inkling of the subject.

All these congruence lattices were generated by using the topos $Quiv$. This is just a small sampling of what can be done with topos theory, within only part of one subject. Topos theory is a logical theory of everything, and its techniques are the most widely applicable.