区间链与序集超幂的完备性
Interval chains and completeness in ultrapowers of ordered sets
摘要：
引入任意序集$T$的超幂$T^{\ast}$作为$T$的无穷小扩张.它作为$T$中序列的等价类的集合而得到，其中对应关系由自然数集合上的超滤子生成。证明了$T^{\ast}$总是满足Cantor性质，同时给出了$T^{\ast}$使$T^{\ast}$完备或满足开完备性的充要条件.也就是说，原始集合的密度决定了扩张的开完备性，而独立地，$T^{\ast}$的完备性由$T$的基数决定。
英文摘要：
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 $T$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
