The lexicographic product of po-groups and $n$-perfect pseudo effect algebras

TL;DR

This paper investigates the Riesz Decomposition Property in lexicographic products of po-groups, especially with integers, and applies findings to classify strong n-perfect pseudo effect algebras via category equivalence.

Contribution

It establishes conditions for Riesz Decomposition Property in lexicographic products and links strong n-perfect pseudo effect algebras to torsion-free directed po-groups.

Findings

Riesz Decomposition Property characterized for lexicographic products

Category of strong n-perfect pseudo effect algebras is equivalent to torsion-free directed po-groups

Results apply specifically to lexicographic products with the integers

Abstract

We will study the existence of different types of the Riesz Decomposition Property for the lexicographic product of two partially ordered groups. A special attention will be paid to the lexicographic product of the group of the integers with an arbitrary po-group. Then we apply these results to the study of $n$-perfect pseudo effect algebras. We show that the category of strong $n$-perfect pseudo-effect algebras is categorically equivalent to the category of torsion-free directed partially ordered groups with RDP$_1.$