Pseudo-BL algebras and pseudo-effect algebras

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 algebra which is non-commutative generalization of weak MV-effect algebra, we investigate the mutual relationship between pseudo-BL algebras and pseudo-effect algebras. We prove that dual pseudo-BL algebra under certain condition can be restricted to a pseudo-weak MV-effect algebra, and a pseudo-weak MV-effect algebra can be extended to dual pseudo-BL algebra under certain condition. Moreover, we give some examples of pseudo-BL algebra which is corresponding to some pseudo-weak MV-effect algebra. Finally, we establish the relationship between pseudo-MV algebras and pseudo-MV effect algebras.