Skip to main content
SpringerLink
Log in

Galois theory for sets of operations closed under permutation, cylindrification, and composition

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

A set of operations on A is shown to be the set of linear term operations of some algebra on A if and only if it is closed under permutation of variables, addition of inessential variables, and composition, and if it contains all projections. A Galois framework is introduced to describe the sets of operations that are closed under the operations mentioned above, not necessarily containing all projections. The dual objects of this Galois connection are systems of pointed multisets, and the Galois closed sets of dual objects are described accordingly. Moreover, the closure systems associated with this Galois connection are shown to be uncountable (even if the closed sets of operations are assumed to contain all projections).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bodnarčuk, V.G., Kalužnin, L.A., Kotov, V.N., Romov, B.A.: Galois theory for Post algebras, I, II. Kibernetika 3, 1–10, 5, 1–9 (1969) (Russian). English translation: Cybernetics 5, 243–252, 531–539 (1969)

  2. Couceiro M., Foldes S.: On closed sets of relational constraints and classes of functions closed under variable substitution. Algebra Universalis 54, 149–165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Couceiro, M., Lehtonen, E.: Classes of operations closed under permutation, cylindrification and composition. In: 40th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2010), pp. 117–121. IEEE Computer Society, Los Alamitos (2010)

  4. Crama Y., Hammer P.L.: Boolean Functions – Theory, Algorithms, and Applications. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  5. Denecke, K., Erné, M., Wismath, S.L. (eds.): Galois Connections and Applications. Kluwer, Dordrecht (2004)

  6. Denecke, K., Wismath, S.L.: Universal Algebra and Applications in Theoretical Computer Science. Chapman & Hall/CRC, Boca Raton (2002)

  7. Everett C.J.: Closure operations and Galois theory in lattices. Trans. Amer. Math. Soc. 55, 514–525 (1944)

    MathSciNet  MATH  Google Scholar 

  8. Geiger D.: Closed systems of functions and predicates. Pacific J. Math. 27, 95–100 (1968)

    MathSciNet  MATH  Google Scholar 

  9. Harnau, W.: Ein verallgemeinerter Relationenbegriff für die Algebra der mehrwertigen Logik, Teil I (Grundlagen), Teil II (Relationenpaare), Teil III (Beweis). Rostock. Math. Kolloq. 28, 5–17 (1985), 31, 11–20 (1987), 32, 15–24 (1987)

  10. Hellerstein L.: On generalized constraints and certificates. Discrete Math. 226, 211–232 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lau D.: Function Algebras on Finite Sets. Springer, Berlin (2006)

    MATH  Google Scholar 

  12. Lehtonen E.: Closed classes of functions, generalized constraints and clusters. Algebra Universalis 63, 203–234 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mal’cev, A.I.: Iterative Post Algebras. Nauka, Novosibirsk (1976) (Russian)

  14. Ore Ø.: Galois connexions. Trans. Amer. Math. Soc. 55, 493–513 (1944)

    MathSciNet  MATH  Google Scholar 

  15. Pippenger N.: Galois theory for minors of finite functions. Discrete Math. 254, 405–419 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pöschel, R.: Concrete representation of algebraic structures and a general Galois theory. In: Kautschitsch, H., Müller, W.B., Nöbauer, W. (eds.) Contributions to General Algebra (Klagenfurt, 1978), pp. 249–272. Johannes Heyn, Klagenfurt (1979)

  17. Pöschel, R.: A general Galois theory for operations and relations and concrete characterization of related algebraic structures. Report R-01/80, Zentralinstitut für Mathematik und Mechanik, Akademie der Wissenschaften der DDR, Berlin (1980)

  18. Pöschel, R., Kalužnin, L.A.: Funktionen- und Relationenalgebren: Ein Kapitel der diskreten Mathematik. Birkhäuser, Basel (1979)

  19. Szabö, L.: Concrete representation of related structures of universal algebras, I. Acta Sci. Math. (Szeged) 40, 175–184 (1978)

    Google Scholar 

  20. Szendrei, Á.: Clones in Universal Algebra. Séminaire de mathématiques supérieures, vol. 99. Les Presses de l’Université de Montréal, Montréal (1986)

Download references

Author information

Authors and Affiliations

  1. Mathematics Research Unit, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Luxembourg

    Miguel Couceiro

  2. Computer Science and Communications Research Unit, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Luxembourg

    Erkko Lehtonen

Authors
  1. Miguel Couceiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erkko Lehtonen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erkko Lehtonen.

Additional information

Presented by R. Poeschel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Couceiro, M., Lehtonen, E. Galois theory for sets of operations closed under permutation, cylindrification, and composition. Algebra Univers. 67, 273–297 (2012). https://doi.org/10.1007/s00012-012-0184-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-012-0184-1

2010 Mathematics Subject Classification

Key words and phrases

Search

Navigation