Archive for Mathematical Logic

, Volume 50, Issue 7–8, pp 713–725 | Cite as

Axiomatizability by \({{\forall}{\exists}!}\)-sentences



A \({\forall\exists!}\)-sentence is a sentence of the form \({\forall x_{1}\cdots x_{n}\exists!y_{1}\cdots y_{m}O(\overline{x},\overline{y})}\), where O is a quantifier-free formula, and \({\exists!}\) stands for “there exist unique”. We prove that if \({\mathcal{C}}\) is (up to isomorphism) a finite class of finite models then \({\mathcal{C}}\) is axiomatizable by a set of \({\forall\exists!}\)-sentences if and only if \({\mathcal{C}}\) is closed under isomorphic images, \({\mathcal{C}}\) has the intersection property, and \({\mathcal{C}}\) is closed under fixed-point submodels. This result is employed to characterize the subclasses of finitely generated discriminator varieties axiomatizable by sentences of the form \({\forall\exists!\bigwedge p=q}\).


Intersection property Preservation theory Elementary class Discriminator variety 

Mathematics Subject Classification (2000)

03C40 03C13 03C07 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Facultad de Matemática, Astronomía y FísicaUniversidad Nacional de CórdobaCórdobaArgentina

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