Structural matrix algebras and their lattices of invariant subspaces

Abstract

A structural matrix algebra R of n x n matrices over a field F has a distributive lattice Lat(R) of invariant subspaces subset of or equal to F-n. This and related known results are reproven here in a fresh way. Further we investigate what happens when R still operates on F-n but is isomorphic to a structural matrix algebra of m x m matrices (m not equal n), Then m < n and Lat(R) contains a certain distributive sublattice but needs not itself be distributive. If in is not too small, a shadow of distributivity, is retained in the form of 2-distributivity and subdirect reducibility of Lat(R). (C) 2004 Elsevier Inc. All rights reserved.