The variety generated by perfect BL-algebras

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Antonio Di Nola
Salvatore Sessa
Francesc Esteva
Lluís Godo
Pere Garcia
Title: The variety generated by perfect BL-algebras: an algebraic approach in a fuzzy setting
Journal: Annals of Mathematics and Artificial Intelligence
Volume 35
Pages: 197-214
Year: 2002


BL-algebras are the Lindembaum 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.