On Realization of Generalized Effect Algebras

A well-known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice L(H)L(H) of all closed subspaces of a separable complex Hilbert space.We show that a generalized effect algebra is representable in the operator generalized effect algebra GD(H)GD(H) of effects of a complex Hilbert space HH iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Riečanová and Zajac. Further, any operator generalized effect algebra GD(H)GD(H) possesses an order determining set of generalized states.