Besides the traditional model of quantum events as projectors in a Hilbert space,
Birkhoff and von Neumann introduced orthomodular lattices as an algebraic generalization.
Later on, more general structures - orthomodular posets and orthoalgebras - were proposed.
All of them generalize Boolean algebras as the event structures of classical systems.
We ask which mathematical properties of Boolean algebras are preserved.
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While Boolean algebras are uniquely determined by their space of probability measures (even two-valued measures, forming the Stone space),
this is by far not the case of orthomodular lattices, whose spaces of probability measures can be arbitrary compact convex spaces.
Finite Boolean algebras are trivially determined by their automorphism groups;
in contrast to this, the automorphism group of an orthomodular lattice can be an arbitrary group.
We have also a positive result:
Like Boolean algebras, orthomodular lattices (and even orthoalgebras) are uniquely determined by the ordering of their poset of Boolean subalgebras.
The reconstruction of the original event structure from its poset of Boolean subalgebras resulted in a new representation of events.