The minus order for projections

Let B(H)Id be the set of all projections on a Hilbert space H. The necessary and sufficient conditions are presented for the existence of the supremum, as well as the infimum, of two arbitrary projections in B(H)Id with respect to the minus order ?. For a projection Q in B(H)Id; the properties of the sets {P : P is an orthogonal projection on H and Q ? P} and {P : P is an orthogonal projection on H and P ? Q} are further explored.