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Pseudo-BL algebras and pseudo-effect algebras. (English) Zbl 1176.03047
Summary: Pseudo-BL algebras and pseudo-effect algebras arose in two rather different fields: fuzzy logics and quantum logics. In this paper, by introducing the notion of pseudo-weak MV-effect algebras, which is a noncommutative generalization of the notion of weak MV-effect algebra, we investigate the mutual relationship between pseudo-BL algebras and pseudo-effect algebras. We prove that a dual pseudo-BL algebra can be restricted under a certain condition to a pseudo-weak MV-effect algebra, and a pseudo-weak MV-effect algebra can be extended to a dual pseudo-BL algebra under a certain condition. Moreover, we give some examples of pseudo-BL algebras which correspond to some pseudo-weak MV-effect algebras. Finally, we establish the relationship between pseudo-MV algebras and pseudo-MV effect algebras.

##### MSC:
 03G25 Other algebras related to logic 03B52 Fuzzy logic; logic of vagueness 03G12 Quantum logic 06D35 MV-algebras
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