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Tabular and Pretabular Varieties of MTL-Algebras

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Relational and Algebraic Methods in Computer Science (RAMiCS 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14787))

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Abstract

Tabular logics have been studied since the sixties, while the notion of pretabular (PT) variety was firstly introduced in the seventies by A.V. Kuznetsov and L. Maksimova, for Heyting algebras. A variety is tabular whenever it is generated by one finite algebra. A variety \(\mathbb {L}\) is PT whenever \(\mathbb {L}\) is not tabular but every variety \(\mathbb {M}\subsetneq \mathbb {L}\) is tabular. In this paper we study the same notion for the case of MTL-algebras. We show that a variety of MTL-algebras is PT if and only if it is generated by each of its infinite chains, and we study some general properties of PT varieties of MTL-algebras. Also, we provide a full classification of tabular and PT varieties of BL and WNM-algebras.

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Notes

  1. 1.

    In [5, Definition 7] the first disjunct of (F) is written incorrectly.

References

  1. Aglianò, P., Montagna, F.: Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181(2–3), 105–129 (2003). https://doi.org/10.1016/S0022-4049(02)00329-8

    Article  MathSciNet  Google Scholar 

  2. Aglianò, P., Ugolini, S.: Strictly join irreducible varieties of residuated lattices. J. Log. Comput. 32(1), 32–64 (2021). https://doi.org/10.1093/logcom/exab059

    Article  MathSciNet  Google Scholar 

  3. Aguzzoli, S., Bianchi, M.: Single chain completeness and some related properties. Fuzzy Sets Syst. 301, 51–63 (2016). https://doi.org/10.1016/j.fss.2016.03.008

    Article  MathSciNet  Google Scholar 

  4. Aguzzoli, S., Bianchi, M.: Minimally many-valued extensions of the monoidal t-norm based logic MTL. In: Petrosino, A., Loia, V., Pedrycz, W. (eds.) WILF 2016. LNCS (LNAI), vol. 10147, pp. 106–115. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-52962-2_9

    Chapter  Google Scholar 

  5. Aguzzoli, S., Bianchi, M.: On linear varieties of MTL-algebras. Soft. Comput. 23(7), 2129–2146 (2019). https://doi.org/10.1007/s00500-018-3423-3

    Article  Google Scholar 

  6. Aguzzoli, S., Bianchi, M.: Strictly join irreducible varieties of BL-algebras: the missing pieces. Fuzzy Sets Syst. 418, 84–100 (2021). https://doi.org/10.1016/j.fss.2020.12.008

    Article  MathSciNet  Google Scholar 

  7. Aguzzoli, S., Bianchi, M.: amalgamation property for some varieties of BL-algebras generated by one finite set of BL-chains with finitely many components. In: Glück, R., Santocanale, L., Winter, M. (eds.) Relational and Algebraic Methods in Computer Science, pp. 1–16. Springer, Cham (2023), https://doi.org/10.1007/978-3-031-28083-2_1

  8. Aguzzoli, S., Bianchi, M., Valota, D.: The classification of all the subvarieties of \(\mathbb{DNMG}\). In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K.T., Krawczak, M. (eds.) IWIFSGN/EUSFLAT -2017. AISC, vol. 641, pp. 12–24. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-66830-7_2

    Chapter  Google Scholar 

  9. Aguzzoli, S., Bova, S., Valota, D.: Free weak nilpotent minimum algebras. Soft. Comput. 21, 79–95 (2017). https://doi.org/10.1007/s00500-016-2340-6

    Article  Google Scholar 

  10. Baker, K.: Finite equational bases for finite algebras in a congruence-distributive equational class. Adv. Math. 24(3), 207–243 (1977). https://doi.org/10.1016/0001-8708(77)90056-1

    Article  MathSciNet  Google Scholar 

  11. Bianchi, M.: Strictly join irreducible elements in the lattice of varieties of BL-algebras. In: 2017 IEEE Symposium Series on Computational Intelligence (SSCI) Proceedings. IEEE (2018). https://doi.org/10.1109/SSCI.2017.8285361

  12. Bianchi, M., Montagna, F.: Supersound many-valued logics and Dedekind-MacNeille completions. Arch. Math. Log. 48(8), 719–736 (2009). https://doi.org/10.1007/s00153-009-0145-3

    Article  MathSciNet  Google Scholar 

  13. Bianchi, M., Montagna, F.: \(n\)-contractive BL-logics. Arch. Math. Log. 50(3–4), 257–285 (2011). https://doi.org/10.1007/s00153-010-0213-8

    Article  MathSciNet  Google Scholar 

  14. Blok, W., Pigozzi, D.: Algebraizable logics, Memoirs of The American Mathematical Society, vol. 77. American Mathematical Society (1989). tinyurl.com/o89ug5o

    Google Scholar 

  15. Blok, W.: Pretabular varieties of modal algebras. Stud. Logica. 39, 101–124 (1980). https://doi.org/10.1007/BF00370315

    Article  MathSciNet  Google Scholar 

  16. Castiglioni, J.L., Zuluaga Botero, W.J.: On finite MTL-algebras that are representable as poset products of archimedean chains. Fuzzy Sets Syst. 382, 57–78 (2020). https://doi.org/10.1016/j.fss.2019.03.015

    Article  MathSciNet  Google Scholar 

  17. Castiglioni, J.L., Zuluaga Botero, W.J.: Split exact sequences of finite MTL-chains. Rev. Un. Mat. Argentina 62(2), 295–304 (2021). https://doi.org/10.33044/revuma.1787

    Article  MathSciNet  Google Scholar 

  18. Cignoli, R., D’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning, Trends in Logic, vol. 7. Kluwer Academic Publishers (1999)

    Google Scholar 

  19. Cintula, P., Esteva, F., Gispert, J., Godo, L., Montagna, F., Noguera, C.: Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann. Pure Appl. Log. 160(1), 53–81 (2009). https://doi.org/10.1016/j.apal.2009.01.012

    Article  MathSciNet  Google Scholar 

  20. Cintula, P., Hájek, P., Noguera, C.: Handbook of Mathematical Fuzzy Logic, vol. 1 and 2. College Publications (2011)

    Google Scholar 

  21. Esakia, L., Meskhi, V.: 5 critical modal systems. Theoria 43, 52–60 (1977)

    Article  MathSciNet  Google Scholar 

  22. Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124(3), 271–288 (2001). https://doi.org/10.1016/S0165-0114(01)00098-7

    Article  MathSciNet  Google Scholar 

  23. Freese, R.S., McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras, Lattices, Varieties: Volume II, vol. 268. American Mathematical Society (2022)

    Google Scholar 

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

    Google Scholar 

  25. Gispert, J.: Axiomatic extensions of the nilpotent minimum logic. Rep. Math. Log. 37, 113–123 (2003). tinyurl.com/nqsle2f

    Google Scholar 

  26. Jenei, S., Montagna, F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Stud. Log. 70(2), 183–192 (2002). https://doi.org/10.1023/A:1015122331293

    Article  MathSciNet  Google Scholar 

  27. Jónsson, B.: Algebras whose congruence lattices are distributive. Math. Scand. 21, 110–121 (1967). https://doi.org/10.7146/math.scand.a-10850

    Article  MathSciNet  Google Scholar 

  28. Katoh, Y., Kowalski, T., Ueda, M.: Almost minimal varieties related to fuzzy logic. Rep. Math. Log. pp. 173–194 (2006). tinyurl.com/h8sc6j3

    Google Scholar 

  29. Kearnes, K., Willard, R.: Residually finite, congruence meet-semidistributive varieties of finite type have a finite residual bound. Proc. Am. Math. Soc. 127(10), 2841–2850 (1999). https://doi.org/10.1090/S0002-9939-99-05097-2

    Article  MathSciNet  Google Scholar 

  30. Kowalski, T.: Pretabular varieties of equivalential algebras. Reports Math. Log. 33, 3–10 (1999)

    MathSciNet  Google Scholar 

  31. Maksimova, L.: Pretabular superintuitionist logic. Alg. Log. 11(5), 308–314 (1972). https://doi.org/10.1007/BF02330744

    Article  Google Scholar 

  32. Maksimova, L.: Pretabular extensions of Lewis s4. Alg. Log. 14(1), 16–33 (1975). https://doi.org/10.1007/BF01668576

    Article  MathSciNet  Google Scholar 

  33. Mardaev, S.I.: Number of prelocally table superintuitionistic propositional logics. Alg. Log. 23(1), 56–66 (1984). https://doi.org/10.1007/BF01979699

    Article  MathSciNet  Google Scholar 

  34. Noguera, C.: Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. thesis, IIIA-CSIC (2006)

    Google Scholar 

  35. Troelstra, A.: On intermediate propositional logics. Indagationes Mathematicae (Proceedings) 68, 141–152 (1965). https://doi.org/10.1016/S1385-7258(65)50019-6

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work has been partially supported by Istituto Nazionale di Alta Matematica (Indam).

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Authors and Affiliations

  1. Department of Computer Science “Giovanni degli Antoni”, Università degli Studi di Milano, Via Celoria 18, 20133, Milano, Italy

    Stefano Aguzzoli & Matteo Bianchi

Authors
  1. Stefano Aguzzoli
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  2. Matteo Bianchi
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Correspondence to Matteo Bianchi .

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Editors and Affiliations

  1. EPITA Research Laboratory (LRE), Rennes, France

    Uli Fahrenberg

  2. Czech Academy of Sciences, Prague, Czech Republic

    Wesley Fussner

  3. Deutsches Zentrum für Luft- und Raumfahrt, Augsburg, Germany

    Roland Glück

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Aguzzoli, S., Bianchi, M. (2024). Tabular and Pretabular Varieties of MTL-Algebras. In: Fahrenberg, U., Fussner, W., Glück, R. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2024. Lecture Notes in Computer Science, vol 14787. Springer, Cham. https://doi.org/10.1007/978-3-031-68279-7_1

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  • DOI: https://doi.org/10.1007/978-3-031-68279-7_1

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