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We give sufficient and necessary conditions to guarantee that a pseudo-effectalgebra admits an $(n+1)$-valued discrete state. We introduce $n$-perfectpseudo-effect algebras as algebras which can be split into $n+1$ comparableslices. We prove that the category of strong $n$-perfect pseudo-effect algebrasis categorically equivalent to the category of torsion-free directed partiallyordered groups of a special type.

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