Computable Scott Sentences for Quasi-Hopfian Finitely Presented Structures

TL;DR

This paper establishes that certain algebraic structures, including some Coxeter groups and the free projective plane, have computable $d$-$\Sigma_2$ Scott sentences, unifying and extending previous results in the field.

Contribution

It proves that all quasi-Hopfian finitely presented structures have $d$-$\Sigma_2$ Scott sentences and extends this to computable structures with specific automorphism conditions, applying to new algebraic examples.

Findings

Every right-angled Coxeter group of finite rank has a computable $d$-$\Sigma_2$ Scott sentence.

Every strongly rigid Coxeter group of finite rank has a computable $d$-$\Sigma_2$ Scott sentence.

The free projective plane of rank 4 has a computable $d$-$\Sigma_2$ Scott sentence.

Abstract

We prove that every quasi-Hopfian finitely presented structure $A$ has a $d$-$\Sigma_2$ Scott sentence, and that if in addition $A$ is computable and $Aut(A)$ satisfies a natural computable condition, then $A$ has a computable $d$-$\Sigma_2$ 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 $d$-$\Sigma_2$ Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable $d$-$\Sigma_2$ 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 $d$-$\Sigma_2$ Scott sentence, thus exhibiting a natural example where the assumption of quasi-Hopfianity is used (since this structure is not Hopfian).