The Classification of All the Subvarieties of \(\mathbb {DNMG}\)

  • Stefano Aguzzoli
  • Matteo BianchiEmail author
  • Diego Valota
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 641)


MTL is the logic of all left-continuous t-norms and their residua. The equivalent algebraic semantics of MTL is constituted by the variety of MTL-algebras, \(\mathbb {MTL}\). The variety \(\mathbb {WNM}\) of weak nilpotent minimum algebras is a major subvariety of \(\mathbb {MTL}\), containing several subvarieties of \(\mathbb {MTL}\) which have been subjects of study in the literature, such as Gödel algebras, Nilpotent Minimum algebras, Drastic Product and Revised Drastic Product algebras, NMG-algebras, as well as Boolean algebras. In this paper we introduce and axiomatise \(\mathbb {DNMG}\), a proper subvariety of \(\mathbb {WNM}\) which contains all the aforementioned varieties. We show that \(\mathbb {DNMG}\) is singly generated by a standard algebra. Further, we determine the structure of the lattice of subvarieties of \(\mathbb {DNMG}\), and we provide the axiomatisation of every subvariety.


WNM-algebras DNMG-algebras NM-algebras Gödel-algebras DP-algebras Axiomatisations of subvarieties Single chain completeness 


  1. 1.
    Aguzzoli, S., Bianchi, M.: On some questions concerning the axiomatisation of WNM-algebras and their subvarieties. Fuzzy Sets Syst. 292, 5–31 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aguzzoli, S., Bianchi, M.: Single chain completeness and some related properties. Fuzzy Sets Syst. 301, 51–63 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aguzzoli, S., Bianchi, M.: Minimally many-valued extensions of the monoidal t-norm based logic MTL. In: Lecture Notes in Artificial Intelligence, vol. 10147, pp. 106–115. Springer (2017)Google Scholar
  4. 4.
    Aguzzoli, S., Bianchi, M., Valota, D.: A note on drastic product logic. In: Information Processing and Management of Uncertainty, Communications in Computer and Information Science, vol. 443, pp. 365–374. Springer (2014)Google Scholar
  5. 5.
    Aguzzoli, S., Bova, S., Valota, D.: Free weak nilpotent minimum algebras. Soft. Comput. 21(1), 79–95 (2017)CrossRefGoogle Scholar
  6. 6.
    Bianchi, M.: The logic of the strongest and the weakest t-norms. Fuzzy Sets Syst. 276, 31–42 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Blok, W., Pigozzi, D.: Algebraizable logics, Memoirs of The American Mathematical Society, vol. 77. American Mathematical Society (1989)Google Scholar
  8. 8.
    Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Universitext, Springer, Heidelberg (2001). Reprint of 1997 ednGoogle Scholar
  9. 9.
    Bova, S., Valota, D.: Finite RDP-algebras: duality. Coproducts Log. J. Log. Comput. 22(3), 417 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cintula, P., Hájek, P., Noguera, C. (eds.): Handbook of Mathematical Fuzzy Logic, vol. 1, 2, 3. College Publications (2011)Google Scholar
  12. 12.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Esteva, F., Godo, L., Noguera, C.: On expansions of WNM t-norm based logics with truth-constants. Fuzzy Sets Syst. 161(3), 347–368 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fodor, J.: Nilpotent minimum and related connectives for fuzzy logic. In: Proceedings of 1995 IEEE International Conference on Fuzzy Systems, pp. 2077–2082. IEEE (1995)Google Scholar
  15. 15.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices: an algebraic glimpse at substructural logics. In: Studies in Logic and The Foundations of Mathematics, vol. 151. Elsevier (2007)Google Scholar
  16. 16.
    García-Cerdaña, À., Noguera, C., Esteva, F.: On the scope of some formulas defining additive connectives in fuzzy logics. Fuzzy Sets Syst. 154(1), 56–75 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gispert, J.: Axiomatic extensions of the nilpotent minimum logic. Rep. Math. Log. 37, 113–123 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hájek, P.: Metamathematics of Fuzzy Logic, Trends in Logic, vol. 4. Kluwer Academic Publishers, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  19. 19.
    Jenei, S.: A note on the ordinal sum theorem and its consequence for the construction of triangular norms. Fuzzy Sets Syst. 126(2), 199–205 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jenei, S., Montagna, F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Stud. Log. 70(2), 183–192 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marchioni, E.: On deductive interpolation for the weak nilpotent minimum logic. Fuzzy Sets Syst. 292, 318–332 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Montagna, F.: Completeness with respect to a chain and universal models in fuzzy logic. Arch. Math. Log. 50(1–2), 161–183 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Noguera, C.: Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. thesis, IIIA-CSIC (2006)Google Scholar
  24. 24.
    Noguera, C., Esteva, F., Gispert, J.: On triangular norm based axiomatic extensions of the weak nilpotent minimum logic. Math. Log. Q. 54(4), 387–409 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, S.: A fuzzy logic for the revised drastic product t-norm. Soft. Comput. 11(6), 585–590 (2007)CrossRefzbMATHGoogle Scholar
  26. 26.
    Wang, S.M., Wang, B.S., Pei, D.W.: A fuzzy logic for an ordinal sum t-norm. Fuzzy Sets Syst. 149(2), 297–307 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Stefano Aguzzoli
    • 1
  • Matteo Bianchi
    • 2
    Email author
  • Diego Valota
    • 1
  1. 1.Department of Computer ScienceUniversità degli Studi di MilanoMilanoItaly
  2. 2.Dipartimento di Scienze Teoriche e ApplicateUniversità degli Studi dell’InsubriaVareseItaly

Personalised recommendations