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

TL;DR

This paper characterizes when pseudo-effect algebras have discrete states with a specific number of values and introduces a class of these algebras called $n$-perfect, establishing a categorical equivalence with certain ordered groups.

Contribution

It provides necessary and sufficient conditions for $(n+1)$-valued states and introduces $n$-perfect pseudo-effect algebras with a categorical equivalence to torsion-free directed partially ordered groups.

Findings

Characterization of $(n+1)$-valued discrete states

Introduction of $n$-perfect pseudo-effect algebras

Categorical equivalence with torsion-free directed groups

Abstract

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.