Skip to main content
SpringerLink
Log in

The product representation theorem for interlaced pre-bilattices: some historical remarks

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

This note aims to unravel the history of the Product Representation Theorem for Interlaced Pre-bilattices. We will see that it has its lattice-theoretic roots in early attempts to solve one of the problems in Birkhoff’s Lattice Theory. The theorem was presented in its full generality by Czédli, Huhn and Szabó at a conference in Szeged, Hungary in 1980 (and published in 1983). This was several years before Ginsberg introduced bilattices at a conference on artificial intelligence in 1986 and in his foundational paper in 1988.

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. Arnold B.H.: Distributive lattices with a third operation defined. Pacific J. Math. 1, 33–41 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avron A.: The structure of interlaced bilattices. Mathematical Structures in Computer Science 6, 287–299 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belnap, N. D.: A useful four-valued logic. Modern uses of multiple-valued logic (Fifth Internat. Sympos., Indiana Univ., Bloomington, Ind., 1975), Episteme, vol. 2, pp. 537 (1977)

  4. Birkhoff G.: Review of ‘S. A. Kiss, Transformations on lattices and structures of logic’ . Bull. Amer. Math. Soc. 54, 675–676 (1948)

    Article  MathSciNet  Google Scholar 

  5. Birkhoff, G.: Lattice Theory, 2nd (revised) edn. Amer. Math. Soc. Colloq. Publ. vol. 21, Providence (1948)

  6. Birkhoff G.: Lattice Theory, 3rd edn. Amer. Math. Soc. Colloq. Publ. vol. 21, Providence (1967)

    MATH  Google Scholar 

  7. Birkhoff G., Kiss S.A.: A ternary operation in distributive lattices. Bull. Amer. Math. Soc. 53, 749–752 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bou F., Jansana R., Rivieccio U.: Varieties of interlaced bilattices. Algebra Universalis 66, 115–141 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bou, F., Rivieccio, U.: The logic of distributive bilattices. Logic Journal of the I.G.P.L. 19, 183–216 (2011)

    Google Scholar 

  10. Czédli, G., Huhn A.P., Szabó, L.: On compatible ordering of lattices. In : Contributions to Lattice Theory (Szeged, 1980), pp. 87–99. Colloq. Math. Soc. J´anos Bolyai 33. North-Holland, Amsterdam (1983)

  11. Czédli, G., Szabó, L.: Quasiorders of lattices versus pairs of congruences. Acta Sci. Math. (Szeged) 60, 207–211 (1995)

    Google Scholar 

  12. Fitting, M.: Bilattices in logic programming. In: Proceedings of the 20th International Symposium on Multiple-Valued Logic, pp. 238–246. The IEEE Computer Society Press, Charlotte (1990)

  13. Fraser, G. A., Horn, A.: Congruence relations in direct products. Proc. Amer. Math. Soc. 26, 390–394 (1970)

    Google Scholar 

  14. Ginsberg, M. L.: Multi-valued logics, in: Proceedings of AAAI-86, Fifth National Conference on Artificial Intelligence, pp. 243–247. Morgan Kaufmann Publishers, Los Altos (1986)

  15. Ginsberg M.L.: Multivalued logics: A uniform approach to inference in artificial intelligence. Comput. Intelligence 4, 265–316 (1988)

    Article  Google Scholar 

  16. Jakubík, J., Kolibiar, M.: On some properties of a pair of lattices. Czech. Math. J. 4, 1–27 (1975) (Russian with English summary)

    Google Scholar 

  17. Jakubík, J.: On lattices whose graphs are isomorphic. Czech. Math. J. 4, 131–142 (1954) (Russian with English summary)

    Google Scholar 

  18. Jakubík, J.: Unoriented graphs of modular lattices. Czech. Math. J. 25(100), 240–246 (1975)

    Google Scholar 

  19. Jung, A., Moshier, M. A.: On the bitopological nature of Stone duality. Technical Report CSR-06-13, School of Computer Science, University of Birmingham (2006)

  20. Kiss, S. A.: Transformations on Lattices and Structures of Logic. New York (1947) See http://babel.hathitrust.org/cgi/pt?id=uc1.b4248825;seq=7;view=1up

  21. Kolibiar, M.: Compatible orderings in semilattices. In: Contributions to General Algebra, 2 (Klagenfurt, 1982), pp. 215–220. Hölder-Pichler-Tempsky, Vienna (1983)

  22. Mobasher B., Pigozzi D., Slutski V., Voutsadakis H.: A duality theory for bilattices. Algebra Universalis 43, 109–25 (2000)

    Google Scholar 

  23. Movsisyan, Y. M., Romanowska, A. B., Smith, J. D. H.: Superproducts, hyperidentities, and algebraic structures of logic programming. J. Combin. Math. Combin. Comput. 58, 101–111 (2006)

    Google Scholar 

  24. Pynko A.P.: Regular bilattices. J. Appl. Non-Classical Logics 10, 93–111 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rivieccio, U.: An Algebraic Study of Bilattice-based Logics. PhD Thesis, University of Barcelona (2010). See http://arxiv.org/abs/1010.2552

  26. Romanowska, A., Trakul, A.: On the structure of some bilattices. In: Halkowska, K., Slawski, B. (eds) Universal and Applied Algebra, pp. 246–253. World Scientific (1989)

  27. Rosenberg, I. G.; Schweigert, D.: Compatible orderings and tolerances of lattices. In: Orders: Description and Roles (L’Arbresle, 1982), North-Holland Math. Stud., vol. 99, pp. 119–150. North-Holland, Amsterdam (1984)

Download references

Author information

Authors and Affiliations

  1. Mathematics and Statistics, La Trobe University, Victoria, 3086, Australia

    Brian A. Davey

Authors
  1. Brian A. Davey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brian A. Davey.

Additional information

Presented by G. Gratzer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Davey, B.A. The product representation theorem for interlaced pre-bilattices: some historical remarks. Algebra Univers. 70, 403–409 (2013). https://doi.org/10.1007/s00012-013-0258-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-013-0258-8

2010 Mathematics Subject Classification

Key words and phrases

Search

Navigation