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A note on implication operators of quantum logic

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Abstract

Qiu (Notes on automata theory based on quantum logic, Sci China Ser F-Inf Sci, 2007, 50(2)154–169) discovered that some basic issues in orthomodular lattice-valued automata rely on bi-implication operator satisfying following condition:

$$ (a_{1}\leftrightarrow b_{1} ) \wedge (a_{2}\leftrightarrow b_{2} )\leq a_{1} \wedge a_{2} \leftrightarrow b_{1} \wedge b_{2} ,~~\forall a_{1},a_{2} ,b_{1},b_{2} \in L, $$

and discovered that bi-implication operator based on Sasaki arrow satisfies this condition if and only if the truth-value lattice L is indeed a Boolean algebra, then asked a question of whether the result is also applied to other four quantum implication operators. We show that the answer is yes, and discuss several other conditions.

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Acknowledgments

The author would like to thank the referees for their very valuable comments and recommendations.

Funding

This project was supported by the National Science Foundation of China (Grant No. 62006168).

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Authors and Affiliations

  1. School of Electronics and Information Engineering, Taizhou University, Taizhou, 318000, China

    Songsong Dai

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  1. Songsong Dai
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Correspondence to Songsong Dai.

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Dai, S. A note on implication operators of quantum logic. Quantum Mach. Intell. 2, 15 (2020). https://doi.org/10.1007/s42484-020-00029-3

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