The meet of incommutable projection operators contradicts Burnside's theorem

Abstract

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.