Riesz Decomposition Properties and the Lexicographic Product of po-groups

TL;DR

This paper investigates conditions under which certain Riesz Decomposition Properties hold in lexicographic products of po-groups, with implications for the structure of pseudo effect algebras and related logical systems.

Contribution

It characterizes when specific RDPs are valid in lexicographic products of po-groups, linking algebraic properties to the structure of pseudo effect algebras.

Findings

Identifies conditions for RDPs in lexicographic po-group products

Connects RDP properties to the structure of pseudo effect algebras

Provides insights into the role of infinitesimal elements in algebraic logic

Abstract

We establish conditions when a certain type of the Riesz Decomposition Property (RDP) holds in the lexicographic product of two po-groups. It is well known that the resulting product is an $\ell$-group if and only if the first one is linearly ordered and the second one is an $\ell$-group. This can be equivalently studied as po-groups with a special type of the RDP. In the paper we study three different types of RDP's. RDP's of the lexicographic products are important for the study of pseudo effect algebras where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas.