Orthocomplemented lattice

Summary

An orthocomplemented lattice, or ortholattice, is a complemented lattice with a chosen complement $a^\perp$ for each element $a$. The map $a \mapsto a^\perp$ must reverse the order and be its own inverse. The main examples are Boolean algebras, with the Boolean complement. Another is the lattice of closed subspaces of a Hilbert space, with the orthogonal complement. The De Morgan laws hold in every ortholattice. Ortholattices are the starting point of the lattice-theoretic approach to quantum logic.