Abstract
In order to provide a unified framework for studying non-commutative algebraic logic, Rump and Yang used three axioms to define quantum B-algebras, which can be seen as implicational subreducts of quantales. Based on the work of Rump and Yang, in this paper we shall continue to investigate the properties of three axioms in quantum B-algebras. First, using two axioms we introduce the concept of generalized quantum B-algebras and prove that the opposite of the category GqBAlg of generalized quantum B-algebras is equivalent to the category LogPQ of logical pre-quantales, but we can not prove that pre-quantales can be used as the injective objects in GqBAlg. Next, we use one axiom to propose the concept of C-algebras and show that a C-algebra is a group if and only if each of its elements is dualizing. Further, by dualizing elements of a C-algebra X, we can define different binary operations on X such that X is a moniod. Finally, we by the Zig–Zag relation discuss some properties of quantum B-algebras.
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Acknowledgements
The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 11971286) and the Fundamental Research Funds for the Central University (GK202101009), and also wish to express their sincere thanks to the anonymous referees for their valuable comments and suggestions which improved the quality of the paper.
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Presented by Constantine Tsinakis; Received October 1, 2018.
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Han, S., Xu, X. A Few Notes on Quantum B-algebras. Stud Logica 109, 1423–1440 (2021). https://doi.org/10.1007/s11225-021-09953-2
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DOI: https://doi.org/10.1007/s11225-021-09953-2