Interval chains and completeness in ultrapowers of ordered sets

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 $T$.