Structural matrix algebras and their lattices of invariant subspaces

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