  We consider the standard quantum logic ${\mathcal L}(H)$ associated to a
complex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ together
with the orthogonal complementation. The orthogonality and compatibility
relations are defined for any logic. In the standard quantum logic, they have a
simple interpretation in terms of operator theory. For example, two closed
subspaces (propositions in the logic ${\mathcal L}(H)$) are compatible if and
only if the projections on these subspaces commute. We present both classical
and more resent results on transformations of ${\mathcal L}(H)$ and the
associated Grassmannians which preserve the orthogonality or compatibility
relation. The first result in this direction was classical Wigner's theorem.
