Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We use this notation to distinguish direct powers from ordinal exponentiation.
References
Aguzzoli, S., Bianchi, M.: On some questions concerning the axiomatisation of WNM-algebras and their subvarieties. Fuzzy Sets Syst. 292, 5–31 (2016)
Aguzzoli, S., Bianchi, M.: Single chain completeness and some related properties. Fuzzy Sets Syst. 301, 51–63 (2016)
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)
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)
Aguzzoli, S., Bova, S., Valota, D.: Free weak nilpotent minimum algebras. Soft. Comput. 21(1), 79–95 (2017)
Bianchi, M.: The logic of the strongest and the weakest t-norms. Fuzzy Sets Syst. 276, 31–42 (2015)
Blok, W., Pigozzi, D.: Algebraizable logics, Memoirs of The American Mathematical Society, vol. 77. American Mathematical Society (1989)
Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Universitext, Springer, Heidelberg (2001). Reprint of 1997 edn
Bova, S., Valota, D.: Finite RDP-algebras: duality. Coproducts Log. J. Log. Comput. 22(3), 417 (2012)
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)
Cintula, P., Hájek, P., Noguera, C. (eds.): Handbook of Mathematical Fuzzy Logic, vol. 1, 2, 3. College Publications (2011)
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)
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)
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)
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)
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)
Gispert, J.: Axiomatic extensions of the nilpotent minimum logic. Rep. Math. Log. 37, 113–123 (2003)
Hájek, P.: Metamathematics of Fuzzy Logic, Trends in Logic, vol. 4. Kluwer Academic Publishers, Dordrecht (1998)
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)
Jenei, S., Montagna, F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Stud. Log. 70(2), 183–192 (2002)
Marchioni, E.: On deductive interpolation for the weak nilpotent minimum logic. Fuzzy Sets Syst. 292, 318–332 (2016)
Montagna, F.: Completeness with respect to a chain and universal models in fuzzy logic. Arch. Math. Log. 50(1–2), 161–183 (2011)
Noguera, C.: Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. thesis, IIIA-CSIC (2006)
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)
Wang, S.: A fuzzy logic for the revised drastic product t-norm. Soft. Comput. 11(6), 585–590 (2007)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Aguzzoli, S., Bianchi, M., Valota, D. (2018). The Classification of All the Subvarieties of \(\mathbb {DNMG}\) . In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 641. Springer, Cham. https://doi.org/10.1007/978-3-319-66830-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-66830-7_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66829-1
Online ISBN: 978-3-319-66830-7
eBook Packages: EngineeringEngineering (R0)