Object preserving congruences of quivers

Let $Q$ be a quiver. Then thin congruences are characterized by the fact that they are fully determined by their object partitions. It would be interesting to consider the dual case of partitions that collapse morphisms but no objects.

Theorem. let $Q$ be a quiver then its lattice of object-preserving congruences $L$ is equal to the direct product of the partition lattices of each of its hom classes: \[ L = \prod_{a,b \in Ob(Q)} Con(Hom(a,b)) \] Proof. let $P$ be a morphism partition for $Q$ and suppose that $a =_P b$ then since this is an object preserving partition, if $s(a) \not= s(b)$ or if $t(a) \not= t(b)$ that would imply that either $s(a) = s(b)$ or that $t(a) = t(b)$ with respect to the object partition induced by $P$, which would be a contradiction of the fact that this is supposed to be an object-preserving partition. So every pair of equal morphisms in $P$ must be parallel in $Q$.

Parallel pairs of morphisms are contained in hom classes $Hom(a,b)$ for pairs of objects $a,b \in Ob(Q)$. It follows that for any morphism $m : a \to b$ the only other morphisms it could be made equal to our those in $Hom(a,b)$. So its part of a congruence lattice $Con(Hom(a,b))$. Then consider all the hom classes of the quiver $Q$, they all have partition lattices that direct product to the set of all object-preserving partitions of $Q$. $\square$

We saw in the theory of thin congruences that the thin congruence associated with any object partition is in fact its equivalence maximal member. So this allows us to characterize the bounds of the lattice of object preserving congruences:

Corollary. let $Q$ be a quiver and let $L$ be its lattice of object-preserving congruences. Then its equivalence minimal member is the trivial partition which equates no elements, and its equivalence maximal member is the thin congruence.

This allows us to better characterize the relationship between thin congruences and object partitions. As we saw here, both thin congruences and object preserving congruences can both be characterized in terms of partition lattices. However, the same is not true for the lattice of congruences $Con(Q)$ of a general quiver $Q$ which need not have any familiar structure in terms of partition lattices.

References:
Quiver in nlab