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) 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 G(D)(H) of effects of a complex Hilbert space H iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Riecanova and Zajac. Further, any operator generalized effect algebra G(D) (H) possesses an order determining set of generalized states.