Effect algebras with the maximality property

TL;DR

This paper explores the maximality property in effect algebras, clarifies its relation to other conditions, and proves that certain effect algebras with specific properties are orthomodular lattices.

Contribution

It establishes the connections between the maximality property and other conditions in effect algebras, and proves new results about effect algebras with Jauch--Piron states.

Findings

Jauch--Piron effect algebra with a countable unital set of states is an orthomodular lattice

A unital set of Jauch--Piron states on an effect algebra with the maximality property is strongly order determining

Various conditions in effect algebras are shown to be stronger than the maximality property

Abstract

The maximality property was introduced in in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in the theory of effect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch--Piron effect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch--Piron states on an effect algebra with the maximality property is strongly order determining.