Algebra universalis

, Volume 68, Issue 1–2, pp 1–16 | Cite as

Implicit definition of the quaternary discriminator

  • Miguel A. Campercholi
  • Diego J. Vaggione


Let A be an algebra. A function f: A n A is implicitly definable by a system of term equations \({\bigwedge t_{i}(x_{1}, . . . , x_{n}, z) = s_{i}(x_{1}, . . . ,x_{n}, z)}\) if f is the only n-ary operation on A making the identities \({t_{i}(\overrightarrow{x}, f(\overrightarrow{x})) \approx s_{i}(\overrightarrow{x}, f(\overrightarrow{x}))}\) hold in A. Let \({\mathcal{K}}\) be a class of non-trivial algebras. We prove that the quaternary discriminator is implicitly definable on every member of \({\mathcal{K}}\) (via the same system) iff \({\mathcal{K}}\) is contained in the class of relatively simple members of some relatively semisimple quasivariety with equationally definable relative principal congruences. As an application, we obtain a characterization of the relatively permutable members of such type of quasivarieties. Furthermore, we prove that every algebra in such a quasivariety has a unique relatively permutable extension.

2010 Mathematics Subject Classification

Primary: 08C15 

Key words and phrases

quaternary discriminator implicit equational definition equationally definable principal congruences 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balbes R., Dwinger P.: Distributive Lattices. University of Missouri Press, Columbia (1974)MATHGoogle Scholar
  2. 2.
    Burris S., Sankappanavar H.: A Course in Universal Algebra. Springer, New York (1981)MATHCrossRefGoogle Scholar
  3. 3.
    Czelakowski J., Dziobiak W.: Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class. Algebra Universalis 27, 128–149 (1990)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Gramaglia H., Vaggione D.: Birkhoff-like sheaf representation for varieties of lattice expansions. Studia Logica 56, 111–131 (1996)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Kollar J.: Congruences and one element subalgebras. Algebra Universalis 9, 266–267 (1979)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Krauss, P., Clark, D.: Global subdirect products. Mem. Amer. Math. Soc. 210 (1979)Google Scholar
  7. 7.
    Pynko, A.: Subquasivarieties of implicative locally-finite quasivarieties. Math. Logic Quarterly (in press)Google Scholar
  8. 8.
    Vaggione D.: Sheaf representation and Chinese remainder theorems. Algebra Universalis 29, 232–272 (1992)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Vaggione D.: Locally Boolean spectra. Algebra Universalis 33, 319–354 (1995)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Vaggione D.: On Jónsson’s theorem. Math. Bohem. 121, 55–58 (1996)MathSciNetMATHGoogle Scholar
  11. 11.
    Vaggione D.: Varieties of shells. Algebra Universalis 36, 483–487 (1996)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Volger H.: Preservation theorems for limits of structures and global sections of sheaves of structures. Math. Z. 166, 27–53 (1970)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Werner H.: Discriminator Algebras, Algebraic Representation and Model Theoretic Properties. Akademie-Verlag, Berlin (1978)MATHGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

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

Personalised recommendations