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

Abstract

It is proved that, for a pre-ordered vector space $X$, the quotient space $(X/A,[W])$ is an Archimedeanization of $X$, where $W$ is the closure of positive wedge $X_{+}$ in ru-topology, $A=W\cap -W$, and $[W]$ is the quotient set of $W$ in $X/A$. A construction of an almost Archimedeanization of a pre-ordered vector space is given. Analogous results are presented for pre-ordered groups. We present an explicit example of a family of vector lattices for which a chain of spaces of infinitely small elements stabilizes first at any chosen ordinal $\alpha ⩽{\omega }_1$.