The Lattice and Simplex Structure of States on Pseudo Effect Algebras

TL;DR

This paper investigates the structure of states, measures, and signed measures on pseudo effect algebras with the Riesz Decomposition Property, revealing their lattice and simplex structures and enabling integral representations.

Contribution

It establishes that the set of Jordan signed measures forms an Abelian Dedekind complete ll-group and characterizes the state space as a Choquet or Bauer simplex, facilitating integral representations.

Findings

Jordan signed measures form an Abelian Dedekind complete ll-group

State space is either empty or a Choquet/Bauer simplex

States can be represented by standard integrals

Abstract

We study states, measures, and signed measures on pseudo effect algebras with some kind of the Riesz Decomposition Property, (RDP). We show that the set of all Jordan signed measures is always an Abelian Dedekind complete $\ell$-group. Therefore, the state space of the pseudo effect algebra with (RDP) is either empty or a nonempty Choquet simplex or even a Bauer simplex. This will allow represent states on pseudo effect algebras by standard integrals.