Maps on idempotent matrices over division rings

Let be an arbitrary division ring and the set of all n×n idempotent matrices over . Under some mild conditions, we give a complete description of maps on that preserve either commutativity, or order, or orthogonality. We give examples showing that our assumptions cannot be relaxed much further. As an application, we will prove a quaternionic analogue of Ovchinnikovʹs result that is important in quantum mechanics. Other applications of our theorems include results on automorphisms of operator and matrix semigroups, local automorphisms, linear preserver problems and geometry of matrices and Grassmannians.