The drastic product * D is known to be the smallest t-norm, since x * D y = 0 whenever x, y < 1. This t-norm is not left-continuous, and hence it does not admit a residuum. So, there are no drastic product t-norm based many-valued logics, in the sense of [7]. However, if we renounce standard completeness, we can study the logic whose semantics is provided by those MTL chains whose monoidal operation is the drastic product. This logic is called S3MTL in [17]. In this note we justify the study of this logic, which we rechristen DP (for drastic product), by means of some interesting properties relating DP and its algebraic semantics to a weakened law of excluded middle, to the Δ projection operator and to discriminator varieties. We shall show that the category of finite DP-algebras is dually equivalent to a category whose objects are multisets of finite chains. This duality allows us to classify all axiomatic extensions of DP, and to compute the free finitely generated DP-algebras.


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  1. 1.
    Baaz, M.: Infinite-valued Gödel logics with 0-1-projections and relativizations. In: Gödel 1996: Logical Foundations of Mathematics, Computer Science and Physics – Kurt Gödel’s Legacy, pp. 23–33. Springer, Berlin (1996)Google Scholar
  2. 2.
    Bianchi, M., Montagna, F.: n-contractive BL-logics. Arch. Math. Log. 50(3-4), 257–285 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Blok, W., Pigozzi, D.: Algebraizable logics. Memoirs of The American Mathematical Society, vol. 77(396). American Mathematical Society (1989)Google Scholar
  4. 4.
    Bova, S., Valota, D.: Finite RDP-algebras: duality, coproducts and logic. J. Log. Comp. 22(3), 417–450 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cintula, P., Hájek, P., Noguera, C.: Handbook of Mathematical Fuzzy Logic, vol. 1 and 2. College Publications (2011)Google Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Esteva, F., Godo, L., Hájek, P., Navara, M.: Residuated fuzzy logics with an involutive negation. Arch. Math. Log. 39(2), 103–124 (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hájek, P.: Metamathematics of fuzzy logic. Trends in Logic, vol. 4. Kluwer Academic Publishers (1998)Google Scholar
  10. 10.
    Horčík, R., Noguera, C., Petrík, M.: On n-contractive fuzzy logics. Math. Log. Q. 53(3), 268–288 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    Johnstone, P.T.: Stone spaces. Cambridge Studies in Advanced Mathematics. Cambridge University Press (1982)Google Scholar
  13. 13.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Trends in Logic, vol. 8. Kluwer Academic Publishers (2000)Google Scholar
  14. 14.
    Kowalski, T.: Semisimplicity, EDPC and Discriminator Varieties of Residuated Lattices. Stud. Log. 77(2), 255–265 (2004)CrossRefzbMATHGoogle Scholar
  15. 15.
    Montagna, F.: Generating the variety of BL-algebras. Soft Comput. 9(12), 869–874 (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    Montagna, F.: Completeness with respect to a chain and universal models in fuzzy logic. Arch. Math. Log. 50(1-2), 161–183 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Noguera, C.: Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. thesis, IIIA-CSIC (2006),
  18. 18.
    Schweizer, B., Sklar, A.: Associative functions and abstract semigroups. Publ. Math. Debrecen 10, 69–81 (1963)MathSciNetGoogle Scholar
  19. 19.
    Wang, S.: A fuzzy logic for the revised drastic product t-norm. Soft Comput. 11(6), 585–590 (2007)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

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

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