Soft Computing

, Volume 21, Issue 1, pp 79–95 | Cite as

Free weak nilpotent minimum algebras

  • Stefano AguzzoliEmail author
  • Simone Bova
  • Diego Valota


We give a combinatorial description of the finitely generated free weak nilpotent minimum algebras and provide explicit constructions of normal forms.



The second author is supported by the FWF Austrian Science Fund (Parameterized Compilation, P26200). The third author is supported by a Marie Curie INdAM-COFUND Outgoing Fellowship.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Aguzzoli S, Bianchi M (2016a) On some questions concerning the axiomatisation of WNM-algebras and their subvarieties. Fuzzy Sets Systems 292:5–31Google Scholar
  2. Aguzzoli S, Bianchi M (2016b) Single chain completeness and some related properties. Fuzzy Sets Systems 301:51–63Google Scholar
  3. Aguzzoli S, Bova S, Gerla B (2012) Free Algebras and Functional Representation. In: Cintula P, Hajek P, Noguera C (eds) Handbook of Mathematical Fuzzy Logic. College Publications, LondonGoogle Scholar
  4. Aguzzoli S, Busaniche M, Marra V (2007) Spectral duality for finitely generated nilpotent minimum algebras, with applications. J Log Comput 17(4):749–765CrossRefzbMATHMathSciNetGoogle Scholar
  5. Aguzzoli S, Gerla B (2008) Normal forms and free algebras for some extensions of MTL. Fuzzy Sets Systems 159(10):1131–1152CrossRefzbMATHMathSciNetGoogle Scholar
  6. Bova S, Valota D (2012) Finite RDP-algebras: duality, coproducts and logic. J Log Comput 22:417–450CrossRefzbMATHMathSciNetGoogle Scholar
  7. Burris S, Sankappanvar H (1981) A Course in Universal Algebra. Springer-Verlag, New YorkCrossRefGoogle Scholar
  8. Busaniche M (2006) Free nilpotent minimum algebras. Math Log Quart 52:219–236CrossRefzbMATHMathSciNetGoogle Scholar
  9. Ciabattoni A, Galatos N, Terui K (2008) From axioms to analytic rules in nonclassical logics pp. 229–240 (2008). In: Proceedings of 23rd IEEE Symposium on Logic in Computer Science (LICS’08)Google Scholar
  10. Esteva F, Godo L (2001) Monoidal t-norm based logic:towards a logic for left-continuous t-norms. Fuzzy Sets Systems 124(3):271–288CrossRefzbMATHMathSciNetGoogle Scholar
  11. Fodor J (1995) Nilpotent minimum and related connectives for fuzzy logic. In: Proceedings of FUZZ-IEEE’95, pp. 2077–2082Google Scholar
  12. Gispert J (2003) Axiomatic extensions of the nilpotent minimum logic. Rep Math Log 37:113–123zbMATHMathSciNetGoogle Scholar
  13. Horn A (1969) Free L-algebras. J Symbol Log 34:475–480CrossRefzbMATHGoogle Scholar
  14. Jenei S, Montagna F (2002) A proof of standard completeness for esteva and godo’s logic MTL. Stud Log 70(2):183–192CrossRefzbMATHMathSciNetGoogle Scholar
  15. Montagna F (2011) Completeness with respect to a chain and universal models in fuzzy logic. Arch Math Log 50:161–183CrossRefzbMATHMathSciNetGoogle Scholar
  16. Noguera C (2007) Algebraic study of axiomatic extensions oftriangular norm based fuzzy logics, vol 27. Monografies de l’Institutd’Investigació en Intel.ligència Artificial, BellaterraGoogle Scholar
  17. Noguera C, Esteva F, Gispert J (2008) On triangular norm based axiomatic extensions of the weak nilpotent minimum logic. Math Log Quart 54(4):387–409CrossRefzbMATHMathSciNetGoogle Scholar
  18. Valota D (2010) Poset representation for free RDP-algebras. In: Hosni H, Montagna F (eds.) Probability, uncertainty and rationality, CRM Series, vol. 10. Edizioni della Scuola Normale Superiore (Pisa 2010)Google Scholar
  19. Wang S (2007) A fuzzy logic for the revised drastic product t-norm. Soft comput 11(6):585–590CrossRefzbMATHGoogle Scholar
  20. Wang S, Wang B, Pei D (2005) A fuzzy logic for anordinal sum t-norm. Fuzzy Sets Systems 149(2):297–307CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversità degli Studi di MilanoMilanoItaly
  2. 2.Institut für Computergraphik und Algorithmen, Technische UniversitätWienAustria
  3. 3.Institut d’Investigació en Intel.ligència Artificial, Campus de la Universitat Autonoma de BarcelonaBellaterraSpain

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