  We study morphisms of the generalized quantum logic of tripotents in
JBW*-triples and von Neumann algebras. Especially, we establish generalization
of celebrated Dye's theorem on orthoisomorphisms between von Neumann lattices
to this new context. We show one-to-one correspondence between maps on
tripotents preserving orthogonality, orthogonal suprema, and reflection $u\to
-u$, on one side, and their extensions to maps that are real linear on sets of
elements with bounded range tripotents on the other side. In a more general
description we show that quantum logic morphisms on tripotent structures are
given by a family of Jordan *-homomorphisms on 2-Peirce subspaces. By examples
we exhibit new phenomena for tripotent morphisms that have no analogy for
projection lattices and demonstrated that the above mention tripotent versions
of Dye's theorem cannot be improved. On the other hand, in a special case of
JBW$^*$-algebras we can generalize Dye's result directly. Besides we show that
structure of tripotents in C$^*$-algebras determines their projection poset and
is a complete Jordan invariant for von Neumann algebras.
