Structural matrix algebras and their lattices of invariant subspaces (Q1765885)

Let \(M_{n}(F)\) be the \(F\)-algebra of all \(n\times n\) matrices over a field \(F\). For any transitive and reflexive binary relation \(\rho\) on \(\left\{ 1,2,\dots,n\right\} \), the set \(M_{n}(F,\rho)\) of all matrices \(\left[ a_{ij}\right] \) in \(M_{n}(F)\) for which \(a_{ij}=0\) for all \((i,j)\notin\rho\) is a subalgebra of \(M_{n}(F)\) called a ``structural matrix algebra''. Structural matrix algebras have been studied (under various names) in papers ranging from \textit{R. M. Thrall} [Can. J. Math. 4, 227--239 (1952; Zbl 0048.02304)] to \textit{J. H. Meyer} and \textit{L. van Wyk} [Chin. J. Math. 24, 251--264 (1996; Zbl 0869.16018)]. The authors give new proofs of some of the earlier results, including the fact that the lattice of \(M_{n}(F,\rho)\)-invariant subspaces of \(F^{n}\) is always distributive, and providing a suitable converse. They also examine the situation where a subalgebra \(\mathcal{R}\) of \(M_{n}(F)\) is isomorphic to some structural matrix algebra \(M_{m}(F,\rho)\) with \(m<n\). In the latter case the lattice of \(\mathcal{R}\)-invariant subspaces of \(F^{n}\) is not necessarily distributive, but it contains a specified sublattice which is distributive.