Boolean deductive systems of BL-algebras. (English) Zbl 1030.03048

Summary: BL-algebras appear as Lindenbaum algebras from many-valued logic and were introduced by P. Hájek [Metamathematics of fuzzy logic. Dordrecht: Kluwer (1998; Zbl 0937.03030)]. In this paper Boolean deductive systems and implicative deductive systems of BL-algebras are defined and studied. The following is proved to be equivalent: (i) a deductive systems \(D\) is implicative, (ii) \(D\) is Boolean, (iii) \(L/D\) is a Boolean algebra. Moreover, a BL-algebra \(L\) contains a proper Boolean deductive systems iff \(L\) is bipartite. Local BL-algebras are also characterized. These results generalize some theorems presented by C. S. Hoo for MV-algebras, which are BL-algebras fulfilling an additional double negation law \(x=x^{**}\).


03G25 Other algebras related to logic
03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
06D35 MV-algebras


Zbl 0937.03030
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