sentences:  Suppose M is a von Neumann algebra and "E", "F" are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order « on the family of projections by "E" « "F" if "E" ~ "F' " ≤ "F".In other words, "E" « "F" if there exists a partial isometry "U" ∈ M such that "U*U" = "E" and "UU*" ≤ "F".For closed subspaces "M" and "N" where projections "PM" and "PN", onto "M" and "N" respectively, are elements of M, "M" « "N" if "PM" « "PN".