The characteristic subspace lattice of a linear transformation (Q739114)

In this paper, the authors analyze basic properties of the lattice of the characteristic subspaces \(\mathrm{Chinv}(J)\) based on a nilpotent Jordan matrix \(J\) for the field \(\mathrm{GF}(2)\). \(\mathrm{Chinv}(J)\) is a lattice with inclusion as order, intersection as meet and linear sum as join. Assuming the Shoda condition for a nilpotent Jordan matrix \(J\), it is shown that \(\mathrm{Chinv}(J)\) is not distributive, but it is modular. Moreover, it is not complemented but finite. At the end, the authors show that \(\mathrm{Chinv}(J)\) is a sublattice of \(\mathrm{Inv}(J)\) (the invariant subspaces) and that there is an anti-isomorphism from \(\mathrm{Chinv}(J)\) to \(\mathrm{Chinv}(J)\). Hence, the considered lattice is self-dual.