Boolean deductive systems of BL-algebras

Abstract

BL-algebras rise as Lindenbaum algebras from many valued logic introduced by Hajek [2]. In this paper Boolean ds and implicative ds of BL-algebras are defined and studied. The following is proved to be equivalent: (i) a ds D is implicative, (ii) D is Boolean, (iii) L/D is a Boolean algebra. Moreover, a BL-algebra L contains a proper Boolean ds iff L is bipartite. Local BL-algebras, too, are characterized. These results generalize some theorems presented in [4], [5], [6] for MV-algebras which are BL-algebras fulfiling an additional double negation law x = x **.