When Lexicographic Product of Two po-Groups has the Riesz Decomposition Property

TL;DR

This paper investigates the conditions under which the lexicographic product of two po-groups satisfies various forms of the Riesz Decomposition Property, which is significant for the structure of related algebraic systems and logical frameworks.

Contribution

It extends known results by identifying conditions for the lexicographic product to satisfy RDP, RDP$_1$, and RDP$_2$, linking these properties to the group's order structure.

Findings

Lexicographic product satisfies RDP under certain conditions.

Conditions for RDP$_1$ are established.

RDP$_2$ implies the group is lattice ordered.

Abstract

We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even RDP$_1$, a stronger type of RDP. We recall that a very strong type of RDP, RDP$_2$, entails that the group is lattice ordered. RDP's of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and 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.