  In this paper we study a new class of transformations on the set of all
Hilbert space effects. This consists of the bijective maps which preserve the
order and zero product in both directions. The main result of the paper gives a
complete description of the structure of those transformations. As applications
we obtain additional new results and some former ones as easy corollaries. In
particular, we obtain the form of the ortho-order automorphisms as well as that
of the sequential automorphisms. In the last paragraph of the paper we show
that these two kinds of automorphisms belong to our class of transformations
even when their domain is the set of all effects in a general von Neumann
algebra.
