  In contrast to conjunctions of commutable projection operators unambiguously
represented by their meets, the mathematical representation of conjunctions of
incommutable projection operators is a question that has yet to be solved. This
question relates to another asking whether the set of the column spaces of the
projection operators, commutable and incommutable alike, forms a lattice. As it
is demonstrated in the paper, if the Hilbert space is finite, the column spaces
of the incommutable projection operators cannot be elements of one partially
ordered set in accordance with Burnside's theorem on matrix algebras.
