Linear Orders的凝聚力
Cohesive Powers of Linear Orders
摘要：
可计算结构的内聚能力可以看作是有效不可分解集合上的有效超积，称为内聚集合。研究了常见的可计算线性序$\mathcal{L}$的凝聚力$Π_{C}% \mathcal{L}$的同构类型。如果$% \mathcal{L}$同构于自然数的有序集$\mathbb{N}$，并且有一个可计算的后继函数，那么$Π_{C}\mathcal{L}$同构于$\mathbb{N}+\mathbb{Q}\times \mathbb{Z}.$这里，$+$代表两个阶的和，$\times $代表两个阶的字典积。我们构造了同构于$\mathbb{N},$的可计算线性序$\mathcal{L}_{1}$和$\mathcal{L}_{2}$，它们都具有不可计算后继函数，使得$Π_{C}\mathcal{L}_{1}\mathbb{\ }$同构于$\mathbb{N}+% \mathbb{Q}\times \mathbb{Z}$，而$Π_{C}\mathcal{L}_{2}$不是$.$。当内聚能力保持所有$Π_{2}^{0}$和$Σ_{2}^{0}$语句时，我们提供了新的$Π_{3}^{0}$语句$Φ$和可计算结构$% \mathcal{M}$的例子，使得$\mathcal{M}\vDash Φ$而$Π_{C}\mathcal{M}% \vDash \urcorner Φ.$
英文摘要：
Cohesive powers of computable structures can be viewed as effective ultraproducts over effectively indecomposable sets called cohesive sets. We investigate the isomorphism types of cohesive powers $Π_{C}% \mathcal{L}$ for familiar computable linear orders $\mathcal{L}$. If $% \mathcal{L}$ is isomorphic to the ordered set of natural numbers $\mathbb{N}$ and has a computable successor function, then $Π_{C}\mathcal{L}$ is isomorphic to $\mathbb{N}+\mathbb{Q}\times \mathbb{Z}.$ Here, $+$ stands for the sum and $\times $ for the lexicographical product of two orders. We construct computable linear orders $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ isomorphic to $\mathbb{N},$ both with noncomputable successor functions, such that $Π_{C}\mathcal{L}_{1}\mathbb{\ }$is isomorphic to $\mathbb{N}+% \mathbb{Q}\times \mathbb{Z}$, while $Π_{C}\mathcal{L}_{2}$ is not$.$ While cohesive powers preserve all $Π_{2}^{0}$ and $Σ_{2}^{0}$ sentences, we provide new examples of $Π_{3}^{0}$ sentences $Φ$ and computable structures $% \mathcal{M}$ such that $\mathcal{M}\vDash Φ$ while $Π_{C}\mathcal{M}% \vDash \urcorner Φ.$
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