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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aglianó, P., and M. Marcos, Varieties of K-lattices, Fuzzy Sets and Systems, 2021, https://doi.org/10.1016/j.fss.2021.08.020.
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.
Barr, M., \(*\)-Autonomous Categories, vol. 752 of Lecture Notes in Mathematics, Springer-Verlag, 1979.
Blount, K., and C. Tsinakis, The Structure of residuated lattices, International Journal of Algebra and Computation 13: 437–461, 2003.
Busaniche, M., and R. Cignoli, Constructive logic with strong negation as a substructural logic, Journal of Logic and Computation 20: 761–793, 2010.
Busaniche, M., and R. Cignoli, Residuated lattices as an algebraic semantics for paraconsistent Nelson logic, Journal of Logic and Computation 19: 1019–1029, 2009.
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.
Busaniche, M., and R. Cignoli, Commutative residuated lattices represented by twist-products, Algebra Universalis 71: 5–22, 2014.
Cabrer, L., and H. Priestley, A general framework for product representations: billatices and beyond, Logic Journal of the IGPL 23 (5): 816–841, 2015.
Cignoli, R., The class of Kleene algebras satisfying an interpolation property and Nelson algebras, Algebra Universalis 23: 262–292, 1986.
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.
Galatos, N., and J.G. Raftery, Adding involution to residuated structures, Studia Logica 77: 181–207, 2004.
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.
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.
Galatos, N., and P. Jipsen, The structure of generalized BI-algebras and weakening relation algebras, Algebra Universalis 81(3): 2020.
Hart, J.B., L. Rafter, and C. Tsinakis, The structure of commutative residuated lattices, International Journal of Algebra and Computation 12: 509–524, 2002.
Kalman, J., Lattices with involution, Transactions of the American Mathematical Society 87: 485–491, 1958.
Kracht, M., On extensions of intermediate logics by strong negation, Journal of Philosophical Logic 27: 49–73, 1998.
Nelson, D., Constructible falsity, Journal of Symbolic Logic 14: 16–26, 1949.
Odintsov, S.P., On the representation of N4-lattices, Studia Logica 76: 385–405, 2004.
Odintsov, S.P., Constructive Negations and Paraconsistency, vol. 26 of Trends in Logic, Springer, 2008
Rasiowa, H., N-lattices and constructive logic with strong negation, Funtamenta Mathematicae 46(1): 61–80, 1968.
Rasiowa, H., An algebraic approach to non-classical logics, North-Holland, 1974.
Rivieccio, U., Implicative twist-structures, Logic Journal of the IGPL 28(5): 973–999, 2020.
Rivieccio, U., and H. Ono, Modal Twist-Structures, Algebra Universalis 71(2): 155–186, 2014.
Rosenthal, K., Quantales and Their Applications, vol. 234 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990.
Sendlewski, A., Nelson algebras through Heyting ones. I, Studia Logica 49: 105–126, 1990.
Spinks, M., and R. Veroff, Constructive logic with strong negation is a substructural logic I, Studia Logica 88: 325–348, 2008.
Spinks, M. and R. Veroff, Constructive logic with strong negation is a substructural logic II, Studia Logica 89: 401–425, 2008.
Tsinakis, C., and A.M. Wille, Minimal Varieties of Involutive Residuated Lattices, Studia Logica 83: 407–423, 2006.
Vakarelov, D., Notes on N-lattices and constructive logic with strong negation, Studia Logica 34: 109–125, 1997.
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
Corresponding author
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
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-022-09988-z