Skip to main content
SpringerLink
Log in

Priestley Duality for Bilattices

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

We develop a Priestley-style duality theory for different classes of algebras having a bilattice reduct. A similar investigation has already been realized by B. Mobasher, D. Pigozzi, G. Slutzki and G. Voutsadakis, but only from an abstract category-theoretic point of view. In the present work we are instead interested in a concrete study of the topological spaces that correspond to bilattices and some related algebras that are obtained through expansions of the algebraic language.

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. Arieli O., Avron A.: ‘Reasoning with logical bilattices’. Journal of Logic, Language and Information 5(1), 25–63 (1996)

    Article  Google Scholar 

  2. Avron A.: ‘The structure of interlaced bilattices’. Mathematical Structures in Computer Science 6(3), 287–299 (1996)

    Article  Google Scholar 

  3. Blok, W. J., and D. Pigozzi, Algebraizable logics, vol. 396 of Mem. Amer. Math. Soc., A.M.S., Providence, 1989.

  4. Bou F., Jansana R., Rivieccio U.: ‘Varieties of interlaced bilattices’. Algebra Universalis 66(1-2), 115–141 (2011)

    Article  Google Scholar 

  5. Bou F., Rivieccio U.: ‘The logic of distributive bilattices’. Logic Journal of the I.G.P.L. 19(1), 183–216 (2011)

    Google Scholar 

  6. Cignoli R., Torrens A.: ‘Free Algebras in Varieties of Glivenko MTL-algebras Satisfying the Equation 2(x 2) = (2x)2′. Studia Logica 83(1-3), 157–181 (2006)

    Article  Google Scholar 

  7. Cornish W.H., Fowler P.R.: ‘Coproducts of De Morgan algebras’. Bull. Austral. Math. Soc. 16(1), 1–13 (1977)

    Article  Google Scholar 

  8. Cornish W.H., Fowler P.R.: ‘Coproducts of Kleene algebras’. J. Austral. Math. Soc. Ser. A 27(2), 209–220 (1979)

    Article  Google Scholar 

  9. Davey B.A.: ‘Dualities for equational classes of Brouwerian algebras and Heyting algebras’. Transactions of the American Mathematical Society 221(1), 119–146 (1976)

    Article  Google Scholar 

  10. Davey B.A., Priestley H.A.: Introduction to Lattices and Order, 2nd edition. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  11. Esakia L.: ‘Topological Kripke models’. Soviet Math. Dokl. 15, 147–151 (1974)

    Google Scholar 

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

  13. Fitting M.: ‘Kleene’s three-valued logics and their children’. Fundamenta Informaticae 20(1-3), 113–131 (1994) Special Anniversary Issue: 20th volume of Fundamenta Informaticae.

    Google Scholar 

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

    Article  Google Scholar 

  15. Jung, A., and M. Andrew Moshier, ‘On the bitopological nature of Stone duality’, Tech. rep., School Computer Science, The University of Birmingham, 2006. Technical Report CSR-06-13.

  16. Mobasher B., Pigozzi D., Slutzki G., Voutsadakis G.: ‘A duality theory for bilattices’. Algebra Universalis 43(2-3), 109–125 (2000)

    Article  Google Scholar 

  17. 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 

  18. Odintsov S.P.: ‘Algebraic semantics for paraconsistent Nelson’s logic’. Journal of Logic and Computation 13(4), 453–468 (2003)

    Article  Google Scholar 

  19. Odintsov S.P.: ‘On the representation of N4-lattices’. Studia Logica 76(3), 385–405 (2004)

    Article  Google Scholar 

  20. Odintsov S.P.: ‘Priestley duality for paraconsistent Nelson’s logic’. Studia Logica 96(1), 65–93 (2010)

    Article  Google Scholar 

  21. Priestley H.A.: ‘Representation of distributive lattices by means of ordered Stone spaces’. Bull. London Math. Soc. 2, 186–190 (1970)

    Article  Google Scholar 

  22. Priestley, H. A., ‘Ordered sets and duality for distributive lattices’, in M. Pouzet, and D. Richard (eds.), Orders: Descriptions and Roles, vol. 23 of Annals of Discrete Mathematics, North-Holland, 1984, pp. 39–60.

  23. Rasiowa H.: An algebraic approach to non-classical logics, vol. 78 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1974)

    Google Scholar 

  24. Rivieccio, U., An Algebraic Study of Bilattice-based Logics, Ph.D. Dissertation, University of Barcelona, 2010. Electronic version available at http://arxiv.org/abs/1010.2552

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    A. Jung

  2. Research Center for Integrated Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan

    U. Rivieccio

Authors
  1. A. Jung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. U. Rivieccio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Jung.

Additional information

In memoriam Leo Esakia

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jung, A., Rivieccio, U. Priestley Duality for Bilattices. Stud Logica 100, 223–252 (2012). https://doi.org/10.1007/s11225-012-9376-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-012-9376-0

Keywords

Search

Navigation