Dual quantum B-algebras

  • Fangfang PanEmail author


Quantum B-algebras as implicational subreducts of quantales were introduced by Rump and Yang. They cover the majority of implicational algebras and provide a unified semantics for a wide class of algebraic logics. Some concepts for quantales survive in the framework of quantum B-algebras. In this paper, we first introduce the concept of dual quantum B-algebras (Girard quantum B-algebras). Next, we prove that every dual quantum B-algebra is a residuated poset and that complete dual quantum B-algebras and dual quantales are equivalent to each other. Further, we consider the construction of Girard quantum B-algebras from dual quantum B-algebras.


Quantale Quantum B-algebra Unital quantum B-algebra Dual quantum B-algebra Girard quantum B-algebra 



I first express my gratitude to the Natural Science Program for Basic Research of Shaanxi Province, China (Grant No. 2017JM1015), and I also would like to thank the referees for some of their comments and suggestions for the improvement of this paper.

Compliance with ethical standards

Conflict of interest

The author declares that their is no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsXi’an University of Posts and TelecommunicationsXi’anPeople’s Republic of China

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