Nontrivial quiver families with splitting and preservation properties

Identify nontrivial families of sub-quivers of the Remmel–Whitney quiver whose arrow partition satisfies the stated splitting and preservation properties, and determine the geometric significance of such families.

Background

The paper observes that its product-rule arguments do not depend specifically on the row-rule sub-quivers. Instead, they apply to any partition of the arrows of the Remmel–Whitney quiver into three classes, provided the partition satisfies two conditions: arrows in the positive and negative classes must compose and split in the prescribed way, and the infusion and diffusion bijections must preserve the negative class while their inverses preserve the positive class.

Such a partition yields a sub-quiver whose associated basis has product rules analogous to those established for the row-rule basis. The unresolved issue is whether there are nontrivial families of sub-quivers satisfying these structural conditions and what geometric meaning those families might have.