  The Hilbert space effect algebra is a fundamental mathematical structure
which is used to describe unsharp quantum measurements in Ludwig's formulation
of quantum mechanics. Each effect represents a quantum (fuzzy) event. The
relation of coexistence plays an important role in this theory, as it expresses
when two quantum events can be measured together by applying a suitable
apparatus. This paper's first goal is to answer a very natural question about
this relation, namely, when two effects are coexistent with exactly the same
effects? The other main aim is to describe all automorphisms of the effect
algebra with respect to the relation of coexistence. In particular, we will see
that they can differ quite a lot from usual standard automorphisms, which
appear for instance in Ludwig's theorem. As a byproduct of our methods we also
strengthen a theorem of Molnar.
