Dye’s theorem for tripotents in von Neumann algebras and JBW$$^*$$-triples

We study morphisms of the generalized quantum logic tripotents in JBW $$^*$$ -triples and von Neumann algebras. Especially, we establish a generalization celebrated Dye’s theorem on orthoisomorphisms between lattices to this new context. show existence one-to-one correspondence following maps: (1) posets preserving reflection $$u\rightarrow - u$$ (2) maps triples that preserve are real linear sets elements with bounded range tripotents. In more general description structure given by family Jordan *-homomorphisms Peirce 2-subspaces. By examples demonstrate optimality results. Besides set partial isometries its order orthogonality relation is complete invariant for