Relative uniform convergence and Archimedean property in pre-ordered vector spaces

TL;DR

The work addresses canonical Archimedeanization for a pre-ordered vector space $X$ without requiring lattice structure. It constructs the Archimedeanization by taking $W$ as the ru-closure of $X_+$ and defining $A$ as $W \cap (-W)$, then passing to the quotient to form $(X/A,[W])$. It proves that $[W]$ is an Archimedean cone in $X/A$ and that this quotient is the Archimedeanization of $X$ with a universal property, extending the Luxemburg–Moore–Veksler paradigm beyond lattices. The paper also provides a constructive, transfinite description of the ru-closure via $(X_+)^{(\alpha)}_{ru}$ and introduces invariants $\alpha(X)$ and $\lambda(X)$ to measure Archimedeanization depth, linking relative uniform convergence to Archimedean structure in general pre-ordered spaces.

Abstract

It is proved that, for a pre-ordered vector space $X$, the quotient space $(X/A,[W])$ is the Archimedeanization of $X$, where $W$ is the closure of the positive wedge $X_+$ in the ru-topology, $A=W\cap(-W)$, and $[W]$ is the quotient set of $W$ in $X/A$.

Theorems & Definitions (18)

• Lemma 3.1
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.1
• Corollary 3.1
• Theorem 3.2