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Varieties of BL-algebras. (English) Zbl 1092.03036
This paper is an overview on the lattice Sub(BL) of the subvarieties of the variety BL of the BL-algebras. The authors describe the most important sublattices of Sub(BL). In Section 3 they deal with the lattice of subvarieties of the join of the varieties of Wajsberg algebras, Gödel algebras and Product algebras. This lattice is fully described in the paper of the first three authors, P. Garcia and the reviewer [Arch. Math. Logic 41, No. 7, 673–685 (2002; Zbl 1023.03060)]. In Section 4 the subvarieties of BL which are generated by a single t-norm algebra are sketched and related results can be found in the paper of the second, third and fourth author [Stud. Log. 76, No. 2, 161–200 (2004; Zbl 1045.03048)]. In Section 5 the authors recall the results of the work of P. Aglianò and the fourth author [J. Pure Appl. Algebra 181, No. 2–3, 105–129 (2003; Zbl 1034.06009)] on the varieties generated by BL-chains which are ordinal sums of finitely many Wajsberg hoops. In Section 6 there is a survey on the theory of local and perfect BL-algebras, whose results are contained in the work of of the first three authors, P. Garcia and the reviewer [Ann. Math. Artif. Intell. 35, No. 1–4, 197–214 (2002; Zbl 1004.03023)].

MSC:
03G25 Other algebras related to logic
06D35 MV-algebras
08B15 Lattices of varieties
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
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