Skip to main content
Log in

Varieties of BL-algebras

  • Focus
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this paper we overview recent results about the lattice of subvarieties of the variety BL of BL-algebras and the equational definition of some families of them.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aglianó P, Ferreirim IMA, Montagna F Basic hoops: an algebriac study of continuous t-norms. Studia Logica (to appear)

  2. Aglianó P, Montagna F (2003) Varieties of BL-algebras I: general properties. J Pure Appl Algebra 181:105–129

    Google Scholar 

  3. Belluce LP, Di Nola A, Lettieri A (1993) Local MV-algebras. Rend Circ Mat Palermo 42:347–361

  4. Blok WJ, Ferreirim IMA (2000) On the structure of hoops. Algebra Universalis 43:233–257

    Google Scholar 

  5. Burris S, Sankappanavar HP (1981) A course in Universal Algebra, Graduate texts in Mathematics, Springer, Berlin Heidelberg New York

  6. Busaniche M (2005) Decomposition of BL-chains. To appear in Algebra Universalis Vd. 52, Number 4, 519–525

  7. Chang CC (1958) Algebraic analysis of many-valued logic. Trans Am Math Soc 88:467–490

    Google Scholar 

  8. Cignoli R, Esteva F, Godo L, Torrens A (2000) Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Computing 4:106–112

    Google Scholar 

  9. Cignoli R, D'Ottaviano IML, Mundici D (2000) Algebraic Foundations of Many-valued Reasoning, Kluwer, Doredrecht

  10. Cignoli R, Torrens A (2000) An algebraic analysis of product logic. Mult Val Logic 5:45–65

  11. Cignoli R, Torrens A (2003) Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic. Arch Math Logic 42:361–370

    Google Scholar 

  12. Di Nola A, Lettieri A (1994) Perfect MV-algebras are categorically equivalent to abelian -groups. Studia Logica 53:417–432

    Google Scholar 

  13. Di Nola A, Lettieri A (1999) Equational characterization of all varities of MV algebras. J Algebra 22:463–474

    Google Scholar 

  14. Di Nola A, Esteva F, Garcia P, Godo L, Sessa S (2002) Subvarieties of BL-algebras generated by single-component chains. Arch Math Logic 41:673–685

    Google Scholar 

  15. Di Nola A, Sessa S, Esteva F, Godo L, Garcia P (2002) The variety generated by perfect BL-algebras: an algebraic approach in a fuzzy logic setting. Ann Math Artif Intell 35:197–214

    Google Scholar 

  16. Esteva F, Godo L, Montagna F (2004) Equational Characterization of the Subvarieties of BL Generated by T-Norm Algebras Studia logica 76, 161–200

  17. Ferreirim IMA (1992) On varieties and quasi varieties of hoops and their reducts. PhD Thesis, University of Illinois

  18. Gispert J (2002) Universal classes of MV-chains with applications to many valued logics. Math Logic Q 48:581–601

    Google Scholar 

  19. Gispert J, Mundici D, Torrens A (1999) Ultraproducts of Z with an Application to Many-Valued Logics. J Algebra 219:214–233

    Google Scholar 

  20. Gottwald S (2001) A Treatise on Many-valued Logics. Studies in logic and computation, Research Studies Press, Baldock

  21. Grigolia RS (1977) Algebraic Analysis of Lukasiewicz-Tarski's n-valued logical systems. In: Wojcicki AR, Malinowski G (eds) Selected papers on Lukasiewicz sentential calculus. Ossolineum, Wroclaw, pp 81–92

  22. Hájek P (1998) Metamathematics of fuzzy logic. In: Trends in logic-studia logica library, vol 4. Kluwer, Dordercht/Boston/London

  23. Hájek P (1998) Basic fuzzy logic and BL-algebras. Soft Computing 2:124–128

    Google Scholar 

  24. Haniková Z (2002) A note on propositional tautologies of individual continuous t-norms vol 12. Neural Netw World (5) 453–460

  25. Hecht T, Katrinak T (1972) Equational classes of relative Stone algebras. Notre Dame J Formal Logic 13:248–254

    Google Scholar 

  26. Komori Y (1981) Super- Łukasiewicz implicational logics. Nagoya Math J 84:1119–133

    Google Scholar 

  27. Laskowski MC, Shashoua YV (2002) A classification of BL-algebras. Fuzzy Sets Syst 131:271–282

    Google Scholar 

  28. Mostert PS, Shields AL (1957) On the structure of semigroups on a compact manifold with boundary. Ann Math 65:117–143

    Google Scholar 

  29. Mundici D (1986) Interpretations of AFC*-algebras in Łukasiewicz sentential calculus. J Funct Anal 65:15–63

    Google Scholar 

  30. Panti G (1999) Varieties of MV algebras. J Appl Non-Classical Logic 9:141–157

    Google Scholar 

  31. Rodríguez AJ, Torrens A (1994) Wajsberg algebras and Post algebras. Studia Logica 53:1–19

    Google Scholar 

  32. Turunen E, Sessa S (2001) Local BL-algebras. Int J Multiple Valued Logic 6:229–249

    Google Scholar 

  33. Turunen E (1999) BL-algebras and fuzzy logic. Mathware and Soft Comput 1:49–61

  34. Turunen E (2001) Boolean deductive systems of BL-algebras. Arch Math Logic 40:467–473

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. Di Nola.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Di Nola, A., Esteva, F., Godo, L. et al. Varieties of BL-algebras. Soft Comput 9, 875–888 (2005). https://doi.org/10.1007/s00500-004-0446-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-004-0446-8

Keywords

Navigation