Pseudo-BL algebras and pseudo-effect algebras

Authors:
Xiao-Hong Zhang;Xue-Shuang Fan
Affiliations:
Department of Mathematics, Ningbo University, Ningbo 315211, China;Department of Mathematics, Ningbo University, Ningbo 315211, China
Venue:
Fuzzy Sets and Systems
Year:
2008

Citing 3
Cited 2

BL-algebras and effect algebras

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Qualitative approximations of fuzzy sets and non-classical three-valued logics (I)

RSKT'10 Proceedings of the 5th international conference on Rough set and knowledge technology

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Abstract

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.