Scott sentences for certain groups

Autor: Vikram Saraph, Julia F. Knight
Rok vydání: 2017
Předmět:
Zdroj: Archive for Mathematical Logic. 57:453-472
ISSN: 1432-0665
0933-5846
DOI: 10.1007/s00153-017-0578-z
Popis: 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 $$\varSigma _3$$ 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- $$\varSigma _2$$ ” (the conjunction of a computable $$\varSigma _2$$ sentence and a computable $$\varPi _2$$ sentence). In [9], this was 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 $$\varSigma _3$$ Scott sentence is best possible, while for others, there is a computable d- $$\varSigma _2$$ Scott sentence.
Databáze: OpenAIRE
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