Associating to each pre-order on the indices 1,...,n the corresponding
structural matrix ring, or incidence algebra, embeds the lattice of n-element
pre-orders into the lattice of n x n matrix rings. Rings within the
order-convex hull of the embedding, i.e. matrix rings that contain the ring of
diagonal matrices, can be viewed as incidence algebras of ideal-valued,
generalized pre-order relations.