论文标题
准杂费有限呈现的结构的可计算的Scott句子
Computable Scott Sentences for Quasi-Hopfian Finitely Presented Structures
论文作者
论文摘要
我们证明，每个准霍普式有限呈现的结构$ a $都有$ d $ - $σ_2$ scott句子，并且如果另外$ a $是可计算的，并且$ aut（a）$满足自然计算条件，则$ a $具有可计算的$ d $ -d $ -d $ -d $σ_2$ scott scott句子。这统一了有限呈现结构的Scott句子的几个已知结果，并用于证明其他未经考虑的代数结构具有可计算的$ d $ - $σ_2$ SCOTT句子。特别是，我们表明，每个有限排名的每个右角Coxeter组都有一个可计算的$ d $ - $σ_2$ SCOTT句子，以及任何有限级的刚性Coxeter组。最后，我们表明，排名$ 4 $的免费投影平面具有可计算的$ d $ - $σ_2$ SCOTT句子，因此显示了一个自然的示例，其中使用了Quasi-hopfianity的假设（因为这种结构不是霍普夫人）。
We prove that every quasi-Hopfian finitely presented structure $A$ has a $d$-$Σ_2$ Scott sentence, and that if in addition $A$ is computable and $Aut(A)$ satisfies a natural computable condition, then $A$ has a computable $d$-$Σ_2$ Scott sentence. This unifies several known results on Scott sentences of finitely presented structures and it is used to prove that other not previously considered algebraic structures of interest have computable $d$-$Σ_2$ Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable $d$-$Σ_2$ Scott sentence, as well as any strongly rigid Coxeter group of finite rank. Finally, we show that the free projective plane of rank $4$ has a computable $d$-$Σ_2$ Scott sentence, thus exhibiting a natural example where the assumption of quasi-Hopfianity is used (since this structure is not Hopfian).
