Abstract
Recent work by Busaniche, Galatos and Marcos introduced a very general twist construction, based on the notion of conucleus, which subsumes most existing approaches. In the present paper we extend this framework one step further, so as to allow us to construct and represent algebras which possess a negation that is not necessarily involutive. Our aim is to capture the main properties of the largest class that admits such a representation, as well as to be able to recover the well-known cases—such as (quasi-)Nelson algebras and (quasi-)N4-lattices—as particular instances of the general construction. We pursue two approaches, one that directly generalizes the classical Rasiowa construction for Nelson algebras, and an alternative one that allows us to study twist-algebras within the theory of residuated lattices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Busaniche, M., and R. Cignoli, Residuated lattices as an algebraic semantics for paraconsistent Nelson’s logic, Journal of Logic and Computation 19(6):1019–1029, 2009.
Busaniche, M., and R. Cignoli, The subvariety of commutative residuated lattices represented by twist-products, Algebra Universalis 71(1):5–22, 2014.
Busaniche, M., N. Galatos, and M.A. Marcos, Twist structures and Nelson conuclei, Studia Logica 110: 949–987, 2022.
Celani, S.A., and U. Rivieccio, Intuitionistic modal algebras, Studia Logica. https://doi.org/10.1007/s11225-023-10065-2
Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, 2007.
Hsieh, A., and J.G. Raftery. A finite model property for \(RMI_{min}\), Mathematical Logic Quarterly 52(6):602–612, 2006
Jakl, T., A. Jung and A. Pultr, Bitopology and four-valued logic, Electronic Notes in Theoretical Computer Science 325:201–219, 2016.
Jung, A., P. Maia and U. Rivieccio, Non-involutive twist-structures, Logic Journal of the IGPL, Special Issue on Recovery Operators and Logics of Formal Consistency and Inconsistencies 28(5):973–999, 2020.
Macnab, D.S., An algebraic study of modal operators on Heyting algebras with applications to topology and sheafification, PhD dissertation, University of Aberdeen, 1976.
Odintsov, S.P., Algebraic semantics for paraconsistent Nelson’s logic, Journal of Logic and Computation 13(4):453–468, 2003.
Odintsov, S.P., On the representation of N4-lattices, Studia Logica 76(3):385–405, 2004.
Rasiowa, H., An Algebraic Approach to Non-Classical Logics, volume 78 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1974.
Rivieccio, U., Fragments of Quasi-Nelson: Residuation, Journal of Applied Non-Classical Logics 33(1):52–119, 2023.
Rivieccio, U., Fragments of quasi-Nelson: the algebraizable core, Logic Journal of the IGPL 30(5):807–839, 2022.
Rivieccio, U., Fragments of quasi-Nelson: two negations, Journal of Applied Logic 7:499–559, 2020.
Rivieccio, U., Quasi-N4-lattices, Soft Computing 26:2671–2688, 2022.
Rivieccio, U., Representation of De Morgan and (semi-)Kleene lattices, Soft Computing 24(12):8685–8716, 2020.
Rivieccio, U., T. Flaminio, and T. Nascimento, On the representation of (weak) nilpotent minimum algebras, in 2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Glasgow, United Kingdom, 2020, pp. 1–8.
Rivieccio, U., and R. Jansana, Quasi-Nelson algebras and fragments, Mathematical Structures in Computer Science 31:257–285, 2021.
Rivieccio, U., and Spinks, M., Quasi-Nelson algebras, Electronic Notes in Theoretical Computer Science 344:169–188, 2019.
Rivieccio, U., and M. Spinks, Quasi-Nelson; or, non-involutive Nelson algebras, in D. Fazio, A. Ledda, and F. Paoli (eds.), Algebraic Perspectives on Substructural Logics (Trends in Logic, 55), Springer, 2020, pp. 133–168.
Spinks, M., U. Rivieccio, and T. Nascimento, Compatibly involutive residuated lattices and the Nelson identity, Soft Computing 23:2297–2320, 2019.
Spinks, M., and R. Veroff, Constructive logic with strong negation is a substructural logic. I, Studia Logica 88:325–348, 2008.
Spinks, M., R. Veroff, Paraconsistent constructive logic with strong negation as a contraction-free relevant logic, in J. Czelakowski (ed.), Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science, vol. 16 of Outstanding Contributions to Logic, Springer, Switzerland, 2018, pp. 323–379.
Acknowledgements
We would like to thank the anonymous referees for carefully reading the paper and pointing out issues that needed to be improved. This research is part of the MOSAIC project financed by the European Union’s Marie Skłodowska-Curie grant No. 101007627. The second author was also supported by Universidad Nacional del Litoral, from the research project CAI+D 50620190100088LI El álgebra como herramienta para el tratamiento de problemas de información, research project PIP 11220200101301CO, CONICET Abordaje algebraico y topológico del estudio de sistemas lógicos and research project PICT 2019-00882 CaToAM: triple abordaje semántico de las lógicas modales multivaluadas.
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
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Rivieccio, U., Busaniche, M. Nelson Conuclei and Nuclei: The Twist Construction Beyond Involutivity. Stud Logica (2024). https://doi.org/10.1007/s11225-023-10088-9
Received:
Published:
DOI: https://doi.org/10.1007/s11225-023-10088-9