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PRESERVATION THEOREMS FOR AE-SENTENCES
Part of:
Model theory
Published online by Cambridge University Press: 10 September 2025
Abstract
An AE-sentence is a sentence of the form 
$\forall x_1\ldots x_n \exists ! z_1\ldots z_m \, \alpha (\mathbf {x},\mathbf {z})$, where 
$\alpha $ is a finite conjunction of equations. Given an arbitrary quasivariety 
$\mathcal {Q}$, we give a purely semantical characterization of when a class 
$\mathcal {K} \subseteq \mathcal {Q}$ with 
$S(\mathcal {K}) = \mathcal {Q}$ is axiomatizable relative to 
$\mathcal {Q}$ by AE-sentences. Along the way, we also characterize axiomatizability by generalized AE-sentences, which are of the form described above, except that both the number of existential quantifiers and of equations in 
$\alpha $ are allowed to be infinite. The article concludes with an analysis of how the main theorems can be improved in the context of finitely generated quasivarieties.
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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