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 $L({\mathcal H})$ 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 ${\mathcal G}_D({\mathcal H})$ of effects of a complex Hilbert space ${\mathcal H}$ 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 ${\mathcal G}_D({\mathcal H})$ possesses an order determining set of generalized states.

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