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The Variety Generated by Perfect BL-Algebras: an Algebraic Approach in a Fuzzy Logic Setting

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Abstract

BL-algebras are the Lindenbaum algebras of the propositional calculus coming from the continuous triangular norms and their residua in the real unit interval. Any BL-algebra is a subdirect product of local (linear) BL-algebras. A local BL-algebra is either locally finite (and hence an MV-algebra) or perfect or peculiar. Here we study extensively perfect BL-algebras characterizing, with a finite scheme of equations, the generated variety. We first establish some results for general BL-algebras, afterwards the variety is studied in detail. All the results are parallel to those ones already existing in the theory of perfect MV-algebras, but these results must be reformulated and reproved in a different way, because the axioms of BL-algebras are obviously weaker than those for MV-algebras.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Salerno, Via Allende, 84081, Baronissi, Salerno, Italy

    Antonio Di Nola

  2. Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università di Napoli, via Monteoliveto, 3, 80134, Napoli, Italy

    Salvatore Sessa

  3. Institut d'Investigació en Intelligència Artificial (IIIA), Consejo Superior de Investigaciones Científicas (CSIC), Campus Universitat Autònoma de Barcelona s/n, 08193, Bellaterra, Spain

    Francesc Esteva, Lluis Godo & Pere Garcia

Authors
  1. Antonio Di Nola
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  2. Salvatore Sessa
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  3. Francesc Esteva
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  4. Lluis Godo
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  5. Pere Garcia
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Di Nola, A., Sessa, S., Esteva, F. et al. The Variety Generated by Perfect BL-Algebras: an Algebraic Approach in a Fuzzy Logic Setting. Annals of Mathematics and Artificial Intelligence 35, 197–214 (2002). https://doi.org/10.1023/A:1014539401842

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