Abstract
The present paper is devoted to study the effect of connected and disconnected rotations of Gödel algebras with operators grounded on directly indecomposable structures. The structures resulting from this construction we will present are nilpotent minimum (with or without negation fixpoint, depending on whether the rotation is connected or disconnected) with special modal operators defined on a directly indecomposable algebra. In this paper we will present a (quasi-)equational definition of these latter structures. Our main results show that directly indecomposable nilpotent minimum algebras (with or without negation fixpoint) with modal operators are fully characterized as connected and disconnected rotations of directly indecomposable Gödel algebras endowed with modal operators.
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Aguzzoli, S., Bova, S., Gerla, B.: Free algebras and functional representation for fuzzy logics. In: Cintula, P., et al. (eds.) Handbook of Mathematical Fuzzy Logic, Chapter IX, vol. 2. Studies in Logic, vol. 38, pp. 713–791. College Publications (2011)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press (2001)
Bou, F., Esteva, F., Godo, L., Rodriguez, R.: On the minimum many-values modal logic over a finite residuated lattice. JL&C 21(5), 739–790 (2011)
Busaniche, M.: Free nilpotent minimum algebras. Math. Logic Quart. 52(3), 219–236 (2006)
Caicedo, X., Rodriguez, R.O.: Standard Gödel modal logics. Stud. Logica 94(2), 189–214 (2010)
Caicedo, X., Rodriguez, R.O.: Bi-modal Gödel logic over \([0, 1]\)-valued Kripke frames. J. Logic Comput. 25(1), 37–55 (2015)
Diaconescu, D., Metcalfe, G., Schnüriger, L.: A real-valued modal logic. Logical Methods Comput. Sci. 14(1), 1–27 (2018)
Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124, 271–288 (2001)
Fitting, M.C.: Many-valued modal logics. Fundam. Informat. 15, 235–254 (1991)
Fitting, M.C.: Many-valued modal logics II. Fundam. Informat. 17, 55–73 (1992)
Flaminio, T., Godo, L., Rodríguez, R.O.: A representation theorem for finite Gödel algebras with operators. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds.) WoLLIC 2019. LNCS, vol. 11541, pp. 223–235. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-662-59533-6_14
Flaminio, T., Godo, L., Menchón, P., Rodriguez, R.O.: Algebras and relational frames for Gödel modal logic and some of its extensions. arXiv:2110.02528. Submitted
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers (1998)
Hájek, P.: On fuzzy modal logics \(S5(\mathscr {C})\). Fuzzy Sets Syst. 161(18), 2389–2396 (2010)
Hansoul, G., Teheux, B.: Extending łukasiewicz logics with a modality: algebraic approach to relational semantics. Stud. Logica 101(3), 505–545 (2013)
Hasimoto, Y.: Heyting algebras with operators. Math. Logic. Quart. 47(2), 187–196 (2001)
Horn, A.: Logic with truth values in a linearly ordered Heyting algebra. J. Symbol. Logic 34, 395–405 (1969)
Jenei, S.: On the structure of rotation invariant semigroups. Archiv. Math. Logic 42, 489–514 (2003)
Ono, H., Rivieccio, U.: Modal twist-structures over residuated lattices. Log. J. IGPL 22(3), 440–457 (2014)
Menchón, P., Rodriguez, R.O.: Twist-structures isomorphic to modal nilpotent minimum algebras. Book of Abstracts of First Meeting Brazil-Colombia in Logic, Bogotá, Colombia, 14–17 December 2021 (2021)
Orłowska, E., Rewitzky, I.: Discrete dualities for Heyting algebras with operators. Fundam. Informat. 81, 275–295 (2007)
Palmigiano, A.: Dualities for intuitionistic modal logics. In: Liber Amicorum for Dick de Jongh, Institute for Logic, Language and Computation, pp. 151–167. University of Amsterdam (2004). http://festschriften.illc.uva.nl/D65/palmigiano.pdf
Priest, G.: Many-valued modal logics: a simple approach. Rev. Symbol. Logic 1(2), 190–2013 (2008)
Vidal, A., Esteva, F., Godo, L.: On modal extensions of product fuzzy logic. J. Logic Comput. 27(1), 299–336 (2017)
Acknowledgments
The authors thank the anonymous referees for their comments. Authors acknowledge partial support by the MOSAIC project (EU H2020-MSCA-RISE-2020 Project 101007627). Flaminio and Godo also acknowledge partial support by the Spanish project PID2019-111544GB-C21 funded by MCIN/AEI/10.13039/501100011033. Menchon acknowledge partial support by argentinean projects PIP 112-20200101301CO (CONICET) and PICT-2019-2019-00882 (ANPCyT). The fourth author wants to acknowledge partial support by the following argentinean projects: PIP 112-20150100412CO (CONICET) and UBA-CyT-20020190100021BA.
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Flaminio, T., Godo, L., Menchón, P., Rodriguez, R.O. (2022). Rotations of Gödel Algebras with Modal Operators. In: Ciucci, D., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2022. Communications in Computer and Information Science, vol 1601. Springer, Cham. https://doi.org/10.1007/978-3-031-08971-8_55
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