Generalized Pseudo-Effect Algebras

We introduce generalized pseudo-effect algebras as a non-commutative version of generalized effect algebras.The importance of the algebras of the latter type in quantum physics is based on the fact that they reflect the inner structure of subsets of the positive cone of the group of self-adjoint operators in a Hilbert space. The new algebras are designed to model subsets of group cones as well; but now the underlying po-group may be chosen arbitrarily, in particular it does not need to be abelian.We raise the question when a generalized pseudo-effect algebra is actually representable in the positive cone of a po-group. We are able to give an affirmative answer for two special cases. Both times a property of Riesz kind is involved, defined for our algebra in a similar manner as known for po-groups.