Some remarks on structural matrix rings and matrices with ideal entries

TL;DR

This paper explores the structure of matrix rings associated with pre-orders, showing how various order relations correspond to specific matrix ring configurations and their intersections.

Contribution

It introduces a framework connecting pre-orders with matrix rings, including ideal-valued relations and conjugates of triangular matrix rings.

Findings

Embedding of pre-order lattice into matrix rings

Characterization of incidence algebras as intersections of conjugates

Representation of linear and partial orders via matrix ring conjugates

Abstract

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.