偏序集族
A Poset Hierarchy
摘要：
本文扩展了一份文件的亚伯拉罕和邦纳概括了著名的豪斯多夫刻画类的分散线性订单。Abraham和Bonnet给出了一个偏序集层次，它描述了一类不具有无限反链的离散偏序集（缩写为FAC，表示有限反链条件）。这里的反链是在不可比性的意义上理解的。我们定义了一个比Abraham和Bonnet更大的偏序集层次，以包括更广泛的一类“散射”偏序集，我们称之为$κ$ -散射。这些偏序集不能嵌入任何顺序，使得对于每两个大小为$ < κ$的子集，一个严格小于另一个，其间有一个元素。如果一个线性序具有这个性质并且有$κ$大小，我们称这个集合为$\qkappa$。只有当$κ^{<κ}=κ$。具有如下性质的偏序：对于每个$a<b$，集合$\{x: a<x<b\}$的大小为$\geq κ$，称为弱$κ$ -稠密，不具有弱$κ$ -稠密子集的偏序称为强$κ$ -散布。我们证明了我们的层次包含所有的强$κ$ -分散的FAC偏序集，并且该层次包含在所有的FAC $κ$ -分散偏序集的类中。此外，我们证明了我们的层次结构实际上是所有$κ$ -良好成立的线性序下的反演，字典和和FAC削弱类的封闭。对于$κ=\aleph_0$，我们的层次结构与Abraham-Bonnet定理的层次结构一致。
英文摘要：
This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. Abraham and Bonnet gave a poset hierarchy that characterised the class of scattered posets which do not have infinite antichains (abbreviated FAC for finite antichain condition). An antichain here is taken in the sense of incomparability. We define a larger poset hierarchy than that of Abraham and Bonnet, to include a broader class of ``scattered'' posets that we call $κ$-scattered. These posets cannot embed any order such that for every two subsets of size $ < κ$, one being strictly less than the other, there is an element in between. If a linear order has this property and has size $κ$ we call this set $\qkappa$. Such a set only exists when $κ^{<κ}=κ$. Partial orders with the property that for every $a<b$ the set $\{x: a<x<b\}$ has size $\geq κ$ are called weakly $κ$-dense, and partial orders that do not have a weakly $κ$-dense subset are called strongly $κ$-scattered. We prove that our hierarchy includes all strongly $κ$-scattered FAC posets, and that the hierarchy is included in the class of all FAC $κ$-scattered posets. In addition, we prove that our hierarchy is in fact the closure of the class of all $κ$-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. For $κ=\aleph_0$ our hierarchy agrees with the one from the Abraham-Bonnet theorem.
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。
