The Variety Generated by Perfect BL-Algebras: an Algebraic Approach in a Fuzzy Logic Setting |
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Authors: | Antonio Di Nola Salvatore Sessa Francesc Esteva Lluis Godo Pere Garcia |
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Affiliation: | (1) Dipartimento di Matematica e Informatica, Università di Salerno, Via Allende, 84081 Baronissi, Salerno, Italy;(2) Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università di Napoli, via Monteoliveto, 3, 80134 Napoli, Italy;(3) Institut d'Investigació en Intelligència Artificial (IIIA), Consejo Superior de Investigaciones Científicas (CSIC), Campus Universitat Autònoma de Barcelona s/n, 08193 Bellaterra, Spain |
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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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