Atomic Effect Algebras with the Riesz Decomposition Property

TL;DR

This paper explores the structure of atomic effect algebras with the Riesz Decomposition Property, establishing their equivalence to MV-effect algebras under certain conditions and examining implications for pseudo-effect algebras and states.

Contribution

It proves that sigma-orthocomplete atomic effect algebras with the Riesz Decomposition Property are MV-effect algebras, linking effect algebra properties with partially ordered groups.

Findings

Atomic effect algebras with the Riesz Decomposition Property are MV-effect algebras.

Results connect effect algebras with partially ordered groups with interpolation.

Applications to pseudo-effect algebras and states are demonstrated.

Abstract

We discuss the relationships between effect algebras with the Riesz Decomposition Property and partially ordered groups with interpolation. We show that any $\sigma$-orthocomplete atomic effect algebra with the Riesz Decomposition Property is an MV-effect algebras, and we apply this result for pseudo-effect algebras and for states.