论linear orders的凝聚力
On cohesive powers of linear orders
摘要：
可计算结构的内聚幂是超幂的有效类似物，其中内聚集扮演超滤子的角色。令$ω$、$ζ$和$η$表示自然数、整数和有理数各自的序类型（当被认为是线性序时）。我们调查的凝聚力可计算的线性订单，特别强调$ω$的可计算副本。如果$\mathcal{L}$是$ω$的一个可计算副本，它在计算上同构于$ω$的通常表示，则$\mathcal{L}$的每个内聚势都有序类型$ω+ ζη$。然而，存在$ω$的可计算副本，不一定在计算上同构于通常的表示，具有基本上不等同于$ω+ ζη$的内聚能力。例如，我们表明，有一个可计算的副本$ω$秩序类型的凝聚力$ω+ η$。我们最一般的结果是，如果$X \subseteq \mathbb{N} \setminus \{0\}$是$Σ_2$集合的布尔组合，被认为是一个有限序类型的集合，则存在具有序类型$ω+ σ(X \cup \{ω+ ζη+ ω^*\})$的内聚能力的$ω$的可计算副本，其中$σ(X \cup \{ω+ ζη+ ω^*\})$表示$X$和序类型$ω+ ζη+ ω^*$中序类型的混洗。此外，如果$X$是有限且非空的，则存在具有有序类型$ω+ σ(X)$的内聚能力的$ω$的可计算副本。
英文摘要：
Cohesive powers of computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let $ω$, $ζ$, and $η$ denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of $ω$. If $\mathcal{L}$ is a computable copy of $ω$ that is computably isomorphic to the usual presentation of $ω$, then every cohesive power of $\mathcal{L}$ has order-type $ω+ ζη$. However, there are computable copies of $ω$, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to $ω+ ζη$. For example, we show that there is a computable copy of $ω$ with a cohesive power of order-type $ω+ η$. Our most general result is that if $X \subseteq \mathbb{N} \setminus \{0\}$ is a Boolean combination of $Σ_2$ sets, thought of as a set of finite order-types, then there is a computable copy of $ω$ with a cohesive power of order-type $ω+ σ(X \cup \{ω+ ζη+ ω^*\})$, where $σ(X \cup \{ω+ ζη+ ω^*\})$ denotes the shuffle of the order-types in $X$ and the order-type $ω+ ζη+ ω^*$. Furthermore, if $X$ is finite and non-empty, then there is a computable copy of $ω$ with a cohesive power of order-type $ω+ σ(X)$.
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