Dimension theory for generalized effect algebras

TL;DR

This paper introduces dimension generalized effect algebras (DGEAs), extending the theory of dimension effect algebras (DEAs) to a broader class with a decomposition into types I, II, and III.

Contribution

It generalizes the concept of dimension effect algebras to DGEAs, including a decomposition theorem analogous to the DEA classification.

Findings

DGEAs decompose into type I, II, and III DGEAs.

The theory extends DEA concepts to a more general setting.

DGEAs retain key properties of effect algebras when a unit element exists.

Abstract

In this paper we define and study dimension generalized effect algebras (DGEAs), i.e., Dedekind orthocomplete and centrally orthocomplete generalized effect algebras equipped with a dimension equivalence relation. Our theory is a bona fide generalization of the theory of dimension effect algebras (DEAs), i.e., it is formulated so that, if a DGEA happens to be an effect algebra (i.e., it has a unit element), then it is a DEA. We prove that a DGEA decomposes into type I, II, and III DGEAS in a manner analogous to the type I/II/III decomposition of a DEA.