Every Boolean algebra and, more generally, every modular ortholattice is an orthomodular lattice. P , the projectors on a Hilbert space. An orthomodular lattice is modular if and only if it does not contain the pentagon with common 0 and 1 as a sublattice. Let L be an orthomodular lattice and a,b,c in L such that any one of them commutes with the other two. Every ortholattice is an orthocomplemented poset.
Every orthomodular lattice is an orthomodular poset.
Every Boolean algebra may be represented by some set algebra Q over an appropriate base set Omega . Let w be a statistical operator on some Hilbert space H . So s w is a state in the above sense. If H has dimension greater than two, then every state is obtained in such a way. Then addition becomes an only partially defined operation.