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. For an ortholattice L , the following are equivalent If a commutes with b , then b commutes with a . 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.