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- W4291877980 abstract "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 $Sigma_3$ 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 $d$-$Sigma_2$ (the conjunction of a computable $Sigma_2$ sentence and a computable $Pi_2$ 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 $Sigma_3$ Scott sentence is best possible, while for others, there is a computable $d$-$Sigma_2$ Scott sentence." @default.
- W4291877980 created "2022-08-16" @default.
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- W4291877980 date "2016-06-20" @default.
- W4291877980 modified "2023-09-28" @default.
- W4291877980 title "Scott sentences for certain groups" @default.
- W4291877980 doi "https://doi.org/10.48550/arxiv.1606.06353" @default.
- W4291877980 hasPublicationYear "2016" @default.
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