Riesz space-valued states on pseudo MV-algebras

TL;DR

This paper introduces Riesz space-valued states on pseudo MV-algebras, generalizing classical states, and explores their properties, extremal states, and the structure of the state space, including conditions for it to be a Choquet or Bauer simplex.

Contribution

It defines and studies the properties of Riesz space-valued states on pseudo MV-algebras, including state-morphisms and extremal states, and analyzes the structure of the state space.

Findings

$(R,1_R)$-states generalize classical states on MV-algebras.

The paper characterizes extremal $(R,1_R)$-states and state-morphisms.

Conditions for the state space to be a Choquet or Bauer simplex are established.

Abstract

We introduce Riesz space-valued states, called $(R,1_R)$-states, on a pseudo MV-algebra, where $R$ is a Riesz space with a fixed strong unit $1_R$. Pseudo MV-algebras are a non-commutative generalization of MV-algebras. Such a Riesz space-valued state is a generalization of usual states on MV-algebras. Any $(R,1_R)$-state is an additive mapping preserving a partial addition in pseudo MV-algebras. Besides we introduce $(R,1_R)$-state-morphisms and extremal $(R,1_R)$-states, and we study relations between them. We study metrical completion of unital $\ell$-groups with respect to an $(R,1_R)$-state. If the unital Riesz space is Dedekind complete, we study when the space of $(R,1_R)$-states is a Choquet simplex or even a Bauer simplex.