Discrete (n + 1)-valued states and n-perfect pseudo-effect algebras

We give sufficient and necessary conditions to guarantee that a pseudo-effect algebra admits an (n + 1)-valued discrete state. We introduce n-perfect pseudo-effect algebras as algebras which can be split into n + 1 comparable slices. We prove that the category of strong n-perfect pseudo-effect algebras is categorically equivalent to the category of torsion-free directed partially ordered groups of a special type.