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.
Categories: math.RT math.FA math.LO