Archive for Mathematical Logic

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

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

Article

Abstract

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 \({\forall\exists!\bigwedge p=q}\).

Keywords

Intersection property Preservation theory Elementary class Discriminator variety 

Mathematics Subject Classification (2000)

03C40 03C13 03C07 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Burris S., Sankappanavar H.P.: A course in universal algebra. Springer, New York-Berlin (1981)MATHGoogle Scholar
  2. 2.
    Burris S., Werner H.: Sheaf constructions and their elementary properties. Trans. Am. Math. Soc. 248(2), 193–202 (1979)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caicedo X.: Implicit connectives of algebraizable logics. Stud. Log. 78(1-2), 155–170 (2004)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Campercholi M., Vaggione D.: Algebraically expandable classes. Algebra Universalis 61(2), 151–186 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chang, C.: Unions of chains of models and direct products of models. Summer Institute for, Symbolic Logic, Cornell University , 141–143 (1957)Google Scholar
  6. 6.
    Cignoli R., Loffredo D’Ottaviano I., Mundici D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logic–Studia Logica Library. 7. Kluwer Academic Publishers, Dordrecht: ix, 231 p., (2000)Google Scholar
  7. 7.
    Jónsson B.: Algebras whose congruence lattices are distributive. Math. Scand. 21, 110–121 (1967)MathSciNetMATHGoogle Scholar
  8. 8.
    McKenzie R., McNulty G., Taylor W.: Algebras, Lattices,Varieties, vol. I, Wadsworth Brooks/Cole, Monterey, California (1987)Google Scholar
  9. 9.
    Rabin M.: Classes of models and sets of sentences with the intersection property. Annales de la Faculté Des sciences de l’Université de Clermont 7, 39–53 (1962)MathSciNetGoogle Scholar
  10. 10.
    Werner H.: Discriminator Algebras Algebraic Representation and Model Theoretic Properties. Akademie Verlag, Berlin (1978)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

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

Personalised recommendations