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Quasi-Nelson algebras and fragments

Published online by Cambridge University Press:  11 May 2021

Umberto Rivieccio Open the ORCID record for Umberto Rivieccio [Opens in a new window]  and
Ramon Jansana
Umberto Rivieccio*
Affiliation:
Departamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Natal, Brazil
Ramon Jansana
Affiliation:
Departament de Filosofia, Universitat de Barcelona, Barcelona, Spain
*
*Corresponding author. Email: umberto.rivieccio@ufrn.br
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Abstract

The variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

Keywords

(quasi-)Nelson algebras(quasi-)Kleene algebrasweakly pseudo-complementedtwist-structures

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Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 3: Logical and Semantic Frameworks with Applications , March 2021 , pp. 257 - 285
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Balbes, R. and Dwinger, P. (1974). Distributive Lattices, Columbia, University of Missouri Press.Google Scholar
Blok, W. J. and Pigozzi, D. (1989). Algebraizable Logics, Providence, A.M.S. CrossRefGoogle Scholar
Blok, W. J. and Pigozzi, D. (1994). On the structure of varieties with equationally definable principal congruences III. Algebra Universalis 32 545–608.CrossRefGoogle Scholar
Celani, S. A., Cabrer, L. M. and Montangie, D. (2009). Representation and duality for Hilbert algebras. Central European Journal of Mathematics 7 (3) 463–478.CrossRefGoogle Scholar
Esteva, F. and Godo, L. (2001). Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124 271–288.CrossRefGoogle Scholar
Galatos, N., Jipsen, P., Kowalski, T. and Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Amsterdam, Elsevier.Google Scholar
Gehrke, M. and Harding, J. (2009). Bounded lattice expansions. Journal of Algebra 238 (1), 345–371.CrossRefGoogle Scholar
Grätzer, G. (1978). General Lattice Theory, New York, Academic Press.CrossRefGoogle Scholar
Idziak, P. M., Słomczyńska, K. and Wroński, A. (2009). Fregean varieties. International Journal of Algebra and Computation 19 595–645.CrossRefGoogle Scholar
Liang, F. and Nascimento, T. (2019). Algebraic semantics for quasi-Nelson logic. Lecture Notes in Computer Science 11541 450–466.CrossRefGoogle Scholar
Nelson, D. (1949). Constructible falsity. Journal of Symbolic Logic 14 16–26.CrossRefGoogle Scholar
Odintsov, S. P. (2003). Algebraic semantics for paraconsistent Nelson’s logic. Journal of Logic and Computation 13 453–468.CrossRefGoogle Scholar
Rasiowa, H. (1974). An Algebraic Approach to Non-classical Logics, Amsterdam, North-Holland.Google Scholar
Rivieccio, U. (2014). Implicative twist-structures. Algebra Universalis 71 (2) 155–186.CrossRefGoogle Scholar
Rivieccio, U. (2020a). Fragments of quasi-Nelson: two negations. Journal of Applied Logic 7 (4) 499–559.Google Scholar
Rivieccio, U. (2020b). Representation of De Morgan and (semi-)Kleene lattices. Soft Computing 24 (12) 8685–8716.CrossRefGoogle Scholar
Rivieccio, U., Flaminio, T. and Nascimento, T. (2020a). On the representation of (weak) nilpotent minimum algebras. In: 2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE, Glasgow, United Kingdom, 2020), 1–8, doi: 10.1109/FUZZ48607.2020.9177641.Google Scholar
Rivieccio, U., Jansana, R. and Nascimento, T. (2020b). Two dualities for weakly pseudo-complemented quasi-Kleene algebras. In: Lesot, M.-J., Vieira, S., Reformat, M., Carvalho, J. P., Wilbik, A., Bouchon-Meunier, B., Yager, R. R (eds.) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2020, Communications in Computer and Information Science, vol. 1239, Springer, 634–653.CrossRefGoogle Scholar
Rivieccio, U. and Spinks, M. (2019). Quasi-Nelson algebras. Electronic Notes in Theoretical Computer Science 344 169–188.CrossRefGoogle Scholar
Rivieccio, U. and Spinks, M. (2020). Quasi-Nelson; or, non-involutive Nelson algebras. In: Fazio, D., Ledda, A. and Paoli, F. (eds.) Algebraic Perspectives on Substructural Logics (Trends in Logic 55), Springer, 133–168.Google Scholar
Sankappanavar, H. P. (1987). Semi-De Morgan algebras. Journal of Symbolic Logic 52 712–724.CrossRefGoogle Scholar
Sendlewski, A. (1991). Topologicality of Kleene algebras with a weak pseudocomplementation over distributive p-algebras. Reports on Mathematical Logic 25 13–56.Google Scholar
Spinks, M., Rivieccio., U. and Nascimento, T. (2019). Compatibly involutive residuated lattices and the Nelson identity. Soft Computing 23 2297–2320.CrossRefGoogle Scholar
Spinks, M. & Veroff, R. (2008a). Constructive logic with strong negation is a substructural logic, I. Studia Logica 88 325–348.CrossRefGoogle Scholar
Spinks, M. & Veroff, R. (2008b). Constructive logic with strong negation is a substructural logic, II. Studia Logica 89 401–425.CrossRefGoogle Scholar