Interval chains and completeness in ultrapowers of ordered sets

TL;DR

This paper introduces ultrapowers of ordered sets as infinitesimal extensions, characterizes their completeness properties, and relates these properties to the original set's density and cardinality.

Contribution

It provides necessary and sufficient conditions for ultrapowers of ordered sets to be complete or open complete, linking these properties to the original set's density and size.

Findings

$T^{*}$ always satisfies Cantor's property.

Completeness of $T^{*}$ depends on the cardinality of $T$.

Open completeness of $T^{*}$ depends on the density of $T$.

Abstract

The ultrapower $T^{\ast}$ of an arbitrary ordered set $T$ is introduced as an infinitesimal extension of $T$. It is obtained as the set of equivalence classes of the sequences in $T$, where the corresponding relation is generated by an ultrafilter on the set of natural numbers. It is established that $T^{\ast}$ always satisfies Cantor's property, while one can give the necessary and sufficient conditions for $T$ so that $T^{\ast}$ would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of $T^{\ast}$ is determined by the cardinality of $T$.