On optimal Scott sentences of finitely generated algebraic structures

Matthew Harrison-Trainor; Meng-Che Ho

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arXiv · 1702.06448

On optimal Scott sentences of finitely generated algebraic structures

Matthew Harrison-Trainor·Meng-Che Ho

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Abstract

Scott showed that for every countable structure $\mathcal{A}$, there is a sentence of the infinitary logic $\mathcal{L}_{\omega_1\omega}$, called a Scott sentence for $\mathcal{A}$, whose models are exactly the isomorphic copies of $\mathcal{A}$. Thus, the least quantifier complexity of a Scott sentence of a structure is an invariant that measures the complexity "describing" the structure. Knight et al.~have studied the Scott sentences of many structures. In particular, Knight and Saraph showed that a finitely generated structure always has a $\Sigma^0_3$ Scott sentence. We give a characterization of the finitely generated structures for whom the $\Sigma^0_3$ Scott sentence is optimal. One application of this result is to give a construction of a finitely generated group where the $\Sigma^0_3$ Scott sentence is optimal.

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BibTeX RIS

Matthew Harrison-Trainor, Meng-Che Ho. 2017-02-21. On optimal Scott sentences of finitely generated algebraic structures. https://arxiv.org/abs/1702.06448

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