On optimal Scott sentences of finitely generated algebraic structures
| Autor: | Matthew Harrison-Trainor, Meng-Che Ho |
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| Rok vydání: | 2018 |
| Předmět: |
Discrete mathematics
Algebraic structure Applied Mathematics General Mathematics 010102 general mathematics Structure (category theory) Characterization (mathematics) 16. Peace & justice 01 natural sciences 010101 applied mathematics Countable set Finitely generated group 0101 mathematics Invariant (mathematics) Infinitary logic Sentence Mathematics |
| Zdroj: | Proceedings of the American Mathematical Society. 146:4473-4485 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/proc/14063 |
| Popis: | Scott showed that for every countable structure A \mathcal {A} , there is a sentence of the infinitary logic L ω 1 ω \mathcal {L}_{\omega _1\omega } , called a Scott sentence for A \mathcal {A} , whose countable models are exactly the isomorphic copies of A \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 Σ 3 0 \Sigma ^0_3 Scott sentence. We give a characterization of the finitely generated structures for which the Σ 3 0 \Sigma ^0_3 Scott sentence is optimal. One application of this result is to give a construction of a finitely generated group where the Σ 3 0 \Sigma ^0_3 Scott sentence is optimal. |
| Databáze: | OpenAIRE |
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