Soft Computing

, Volume 9, Issue 12, pp 869–874 | Cite as

Generating the variety of BL-algebras

  • F. Montagna


This paper collects some results from [AFM] and from [AM]. Our purpose is to illustrate some interesting classes of algebras which generate the whole variety of BL-algebras. In particular, we prove that such variety is generated by its finite members and by the class of finite ordinal sums of Lukasiewicz t-norm algebras. Finally, we characterize the BL-chains which generate the whole variety of BL-algebras.


Mathematical Logic Control Engineer Computing Methodology Interesting Class Finite Member 
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  1. 1.
    Aglianó, P., Ferreirim, I.M.A., Montagna, F. (2001) Basic hoops: an algebraic study of continuous t-norms, (preprint)Google Scholar
  2. 2.
    Aglianó, P., Montagna, F. (2003) Varieties of BL-algebras I: general properties. J Pure Appl Algebra 181:105–129Google Scholar
  3. 3.
    Aglianò, P., Ursini, A. (1997) On subtractive varieties III: From ideals to congruences. Algebra Universalis 37:296–333Google Scholar
  4. 4.
    Blok, W.J., Ferreirim, I.M.A. (2000) On the structure of hoops. Algebra Universalis 43:233–257Google Scholar
  5. 5.
    Blok, W., Pigozzi, D. (1989) Algebraizable Logics. Mem Am Math Soc 77:396Google Scholar
  6. 6.
    Busaniche, M. (2002) Decompositions of BL-chains, Preprint University of Buenos AiresGoogle Scholar
  7. 7.
    Chang, C.C. (1958) Algebraic analysis of: many-valued logic. Trans Am Math Soc 88: 467–490Google Scholar
  8. 8.
    Cignoli, R., Esteva, F., Godo, L., Torrens, A. (2000) Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput 4:106–112Google Scholar
  9. 9.
    Cignoli, R., Mundici, D., D'Ottaviano, I.M.L. (2000) Algebraic foundations of many-valued reasoning. Kluwer, DorderchtGoogle Scholar
  10. 10.
    Ferreirim, I.M.A. (1992) On varieties and quasi varieties of hoops and their reducts. PhD Thesis, University of Illinois at ChicagoGoogle Scholar
  11. 11.
    Gumm, P., Ursini, A. (1984) Ideals in Universal Algebra. Algebra Universalis 19:45–54Google Scholar
  12. 12.
    Hájek, P. (1998) Metamathematics of Fuzzy Logic. Trends in Logic, Studia Logica Library no. 4. Kluwer Dordercht/ Boston/ LondonGoogle Scholar
  13. 13.
    Harrop, R. (1958) On the existence of finite models and decision procedures for propositional calculi. Proc Cambridge Philos Soc 58:1–13Google Scholar
  14. 14.
    Laskowski, L.C., Shashoua, Y.V. (2002) A classification of BL-algebras. Fuzzy Sets Sys 131:271–282Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Dipartamento de MatematicaUniversity of SienaSienaItaly

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