The witness set of coexistence of quantum effects and its preservers

‎One of unsolved problems in quantum‎ ‎measurement theory is to characterize coexistence of quantum effects‎. ‎In < this > paper‎, ‎applying positive operator matrix theory‎, ‎we give a mathematical characterization of the witness set of coexistence of‎ ‎quantum effects and obtain a series of properties of coexistence‎. ‎ < We also devote to characterizing bijective morphisms on quantum effects leaving the witness set invariant. > Furthermore‎, ‎applying‎ ‎linear maps preserving commutativity‎, ‎which are characterized by Choi‎, ‎Jafarian and Radjavi [Linear maps preserving commutativity‎, ‎Linear Algebra Appl‎. ‎87 (1987)‎, ‎227--241.]‎, ‎we classify multiplicative general morphisms leaving the witness set invariant on finite dimensional.