某些群体的斯科特句子
Scott sentences for certain groups
摘要：
我们为某些可计算的组给出了Scott语句，并且我们使用索引集计算作为检查我们的Scott语句尽可能简单的一种方式。我们考虑有限秩的有限生成群和无挠阿贝尔群。对于这两类群，可计算群都有可计算$Σ_3$ Scott语句。有时我们可以做得更好。事实上，我们所研究的可计算的可有限生成群都具有“可计算的$d$ - $Σ_2$“（可计算的$Σ_2$语句和可计算的$Π_2$语句的合取）的Scott语句。这已经显示了对于Windows生成的自由群。在这里，我们证明了它对所有的n-生成的阿贝尔群，并为无限二面体群。在有限秩的可计算无挠阿贝尔群中，我们重点研究了秩为$1$的可计算无挠阿贝尔群。这些恰好是$\mathbb{Q}$的加法子群。我们发现，对于这些群体中的一些，可计算的$Σ_3$斯科特句子是最好的可能，而对于其他人，有一个可计算的$d$ - $Σ_2$斯科特句子。
英文摘要：
We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable $Σ_3$ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are "computable $d$-$Σ_2$" (the conjunction of a computable $Σ_2$ sentence and a computable $Π_2$ sentence). This was already shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank $1$. These are exactly the additive subgroups of $\mathbb{Q}$. We show that for some of these groups, the computable $Σ_3$ Scott sentence is best possible, while for others, there is a computable $d$-$Σ_2$ Scott sentence.
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