Skip to main content
SpringerLink
Log in

Twist Structures and Nelson Conuclei

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

Motivated by Kalman residuated lattices, Nelson residuated lattices and Nelson paraconsistent residuated lattices, we provide a natural common generalization of them. Nelson conucleus algebras unify these examples and further extend them to the non-commutative setting. We study their structure, establish a representation theorem for them in terms of twist structures and conuclei that results in a categorical adjunction, and explore situations where the representation is actually an isomorphism. In the latter case, the adjunction is elevated to a categorical equivalence. By applying this representation to the original motivating special cases we bring to the surface their underlying similarities.

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. Aglianó, P., and M. Marcos, Varieties of K-lattices, Fuzzy Sets and Systems, 2021, https://doi.org/10.1016/j.fss.2021.08.020.

    Article  Google Scholar 

  2. Aguzzoli, S., M. Busaniche, B. Gerla, and M. Marcos, On the category of Nelson paraconsistent lattices, Journal of Logic and Computation 27: 2227–2250, 2017.

    Article  Google Scholar 

  3. Barr, M., \(*\)-Autonomous Categories, vol. 752 of Lecture Notes in Mathematics, Springer-Verlag, 1979.

  4. Blount,  K., and C. Tsinakis, The Structure of residuated lattices, International Journal of Algebra and Computation 13: 437–461, 2003.

    Article  Google Scholar 

  5. Busaniche, M., and R. Cignoli, Constructive logic with strong negation as a substructural logic, Journal of Logic and Computation 20: 761–793, 2010.

    Article  Google Scholar 

  6. Busaniche, M., and R. Cignoli, Residuated lattices as an algebraic semantics for paraconsistent Nelson logic, Journal of Logic and Computation 19: 1019–1029, 2009.

    Article  Google Scholar 

  7. Busaniche, M., and R. Cignoli, Remarks on an algebraic semantics for paraconsistent Nelson’s logic, Manuscrito, Center of Logic, Epistemology and the History of Science 34: 99–114, 2011.

    Google Scholar 

  8. Busaniche, M., and R. Cignoli, Commutative residuated lattices represented by twist-products, Algebra Universalis 71: 5–22, 2014.

    Article  Google Scholar 

  9. Cabrer, L., and H. Priestley, A general framework for product representations: billatices and beyond, Logic Journal of the IGPL 23 (5): 816–841, 2015.

    Google Scholar 

  10. Cignoli, R., The class of Kleene algebras satisfying an interpolation property and Nelson algebras, Algebra Universalis 23: 262–292, 1986.

    Article  Google Scholar 

  11. Fidel, M.M., An algebraic study of a propositional system of Nelson, in A.I. Arruda, N.C.A. da Costa, and R. Chuaqui, (eds.), Mathematical Logic. Proceedings of the First Brazilian Conference, vol. 39 of Lectures in Pure and Applied Mathematics, Marcel Dekker Inc, 1978, pp. 99–117.

  12. Galatos, N., and J.G. Raftery, Adding involution to residuated structures, Studia Logica 77: 181–207, 2004.

    Article  Google Scholar 

  13. Galatos, N., and J.G. Raftery, Idempotent residuated structures: some category equivalences and their applications, Transactions of the American Mathematical Society 367(5): 3189–3223, 2015.

    Article  Google Scholar 

  14. Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151 of Studies in Logics and the Foundations of Mathematics, Elsevier, 2007.

  15. Galatos, N., and P. Jipsen, The structure of generalized BI-algebras and weakening relation algebras, Algebra Universalis 81(3): 2020.

  16. Hart, J.B., L. Rafter, and C. Tsinakis, The structure of commutative residuated lattices, International Journal of Algebra and Computation 12: 509–524, 2002.

    Article  Google Scholar 

  17. Kalman, J., Lattices with involution, Transactions of the American Mathematical Society 87: 485–491, 1958.

    Article  Google Scholar 

  18. Kracht, M., On extensions of intermediate logics by strong negation, Journal of Philosophical Logic 27: 49–73, 1998.

    Article  Google Scholar 

  19. Nelson, D., Constructible falsity, Journal of Symbolic Logic 14: 16–26, 1949.

    Article  Google Scholar 

  20. Odintsov, S.P., On the representation of N4-lattices, Studia Logica 76: 385–405, 2004.

    Article  Google Scholar 

  21. Odintsov, S.P., Constructive Negations and Paraconsistency, vol. 26 of Trends in Logic, Springer, 2008

  22. Rasiowa, H., N-lattices and constructive logic with strong negation, Funtamenta Mathematicae 46(1): 61–80, 1968.

    Article  Google Scholar 

  23. Rasiowa, H., An algebraic approach to non-classical logics, North-Holland, 1974.

  24. Rivieccio, U., Implicative twist-structures, Logic Journal of the IGPL 28(5): 973–999, 2020.

    Article  Google Scholar 

  25. Rivieccio, U., and H. Ono, Modal Twist-Structures, Algebra Universalis 71(2): 155–186, 2014.

    Article  Google Scholar 

  26. Rosenthal, K., Quantales and Their Applications, vol. 234 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990.

  27. Sendlewski, A., Nelson algebras through Heyting ones. I, Studia Logica 49: 105–126, 1990.

    Article  Google Scholar 

  28. Spinks, M., and R. Veroff, Constructive logic with strong negation is a substructural logic I, Studia Logica 88: 325–348, 2008.

    Article  Google Scholar 

  29. Spinks, M. and R. Veroff, Constructive logic with strong negation is a substructural logic II, Studia Logica 89: 401–425, 2008.

    Article  Google Scholar 

  30. Tsinakis, C., and A.M. Wille, Minimal Varieties of Involutive Residuated Lattices, Studia Logica 83: 407–423, 2006.

    Article  Google Scholar 

  31. Vakarelov, D., Notes on N-lattices and constructive logic with strong negation, Studia Logica 34: 109–125, 1997.

    Google Scholar 

Download references

Acknowledgements

The results of this paper are supported by the following research projects: CAI+D 50620190100088LI—El álgebra como herramienta para el tratamiento de problemas de información, founded by Universidad Nacional del Litoral (Busaniche and Marcos). PICT 2019-00882—CaToAM: triple abordaje semántico de las lógicas modales multivaluadas, founded by Agencia Nacional de Promoción de la Investigación, el Desarrollo Tecnológico y la Innovación, Argentina (Busaniche and Marcos). We are grateful with the referees that read the first version of the paper and pointed out some related bibliography. We would also like to thank Adam P\(\breve{r}\)enosil and Umberto Rivieccio for their comments and suggestions.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ingeniería Química, CONICET-UNL, Santiago del Estero 2829, Santa Fe, Argentina

    Manuela Busaniche & Miguel Andrés Marcos

  2. University of Denver, 2390 S. York St., Denver, USA

    Nikolaos Galatos

Authors
  1. Manuela Busaniche
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nikolaos Galatos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miguel Andrés Marcos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuela Busaniche.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Presented by Francesco Paoli

The third author name was incorrectly abbreviated as M.A.E. Marcos. It is corrected as M.A Marcos.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Busaniche, M., Galatos, N. & Marcos, M.A. Twist Structures and Nelson Conuclei. Stud Logica 110, 949–987 (2022). https://doi.org/10.1007/s11225-022-09988-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-022-09988-z

Keywords

Search

Navigation