Abstract
A structural matrix algebra R of n × n matrices over a field F has a distributive lattice Lat ( R ) of invariant subspaces ⊆ 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 × 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 ) .
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