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 Rie\v{c}anov'a and Zajac. Further, any operator generalized effect algebra ${\mathcal G}_D({\mathcal H})$ possesses an order determining set of generalized states.