  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. Certain conjugates of the upper or lower
triangular matrix rings correspond to the various linear orderings of the
indices, and the incidence algebras of partial orderings arise as intersections
of such conjugate matrix rings.
