Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scott sentences for certain groups   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.

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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex