It is not complete, since sup N 0 does not exist. The smallest and greatest elements of this poset are 0 and 1 respectively. Yet P can be turned into a poset the same way as before. This poset turns out to be even complete. One such substructure is that of posets.
Every Boolean algebra is trivially a lattice.
The Standard Quantum Logic P forms a lattice. Every lattice can be turned into a poset by the method already mentioned. A lattice is modular if and only if it does not contain the pentagon as an isomorphic sublattice. Every Boolean algebra is an ortholattice. Let L be some ortholattice.