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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 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 lineardoi:10.18514/mmn.2011.329 fatcat:d4b5tdyd3regrpo3xdhrnrpud4