Homogeneous orthocomplete effect algebras are covered by MV-algebras

TL;DR

This paper investigates the structure of homogeneous effect algebras, showing that under certain conditions they are covered by MV-algebras, thus linking effect algebra properties with lattice and MV-algebra structures.

Contribution

It demonstrates that homogeneous orthocomplete effect algebras are covered by MV-algebras, extending the understanding of their lattice and effect algebra structures.

Findings

Every block of an Archimedean homogeneous effect algebra satisfying (W+) is lattice ordered.

Homogeneous orthocomplete effect algebras have lattice-ordered blocks.

Finite homogeneous effect algebras are covered by MV-algebras.

Abstract

The aim of our paper is twofold. First, we thoroughly study the set of meager elements Mea(E) and the set of hypermeager elements HMea(E) in the setting of homogeneous effect algebras E. Second, we study the property (W+) and the maximality property introduced by Tkadlec as common generalizations of orthocomplete and lattice effect algebras. We show that every block of an Archimedean homogeneous effect algebra satisfying the property (W+) is lattice ordered. Hence such effect algebras can be covered by ranges of observables. As a corollary, this yields that every block of a homogeneous orthocomplete effect algebra is lattice ordered. Therefore finite homogeneous effect algebras are covered by MV-algebras.