Type-Decomposition of a Pseudo-Effect Algebra

TL;DR

This paper extends the theory of effect algebra decomposition to pseudo-effect algebras, introducing a type-determining set concept that enables their decomposition into direct summands.

Contribution

It generalizes the notion of type-determining sets to pseudo-effect algebras and develops their decomposition theory.

Findings

Centrally orthocomplete PEAs can be decomposed into direct summands.

Type-determining sets induce natural decompositions of PEAs.

The theory extends effect algebra results to noncommutative cases.

Abstract

The theory of direct decomposition of a centrally orthocomplete effect algebra into direct summands of various types utilizes the notion of a type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly) noncommutative version of an effect algebra. In this article we develop the basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set to PEAs, and show that TD sets induce decompositions of centrally orthocomplete PEAs into direct summands.