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Preorders, Partial Semigroups, and Quantales

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Relational and Algebraic Methods in Computer Science (RAMiCS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12062))

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Abstract

It is known that each powerset quantale is embeddable into some relational unital quantale whose underlying set is the powerset of some preorder. An aim of this paper is to understand the relational embedding as a relationship between quantales and preorders. For that, this paper introduces the notion of weak preorders, a functor from the category of weak preorders to the category of partial semigroups, and a functor from the category of partial semigroups to the category of quantales and lax homomorphisms. By using these two functors, this paper shows a correspondence among four classes of weak preorders (including the class of ordinary preorders), four classes of partial semigroups, and four classes of quantales. As a corollary of the correspondence, we can understand the relational embedding map as a natural transformation between functors onto certain category of quantales.

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Acknowledgements

The authors thank Izumi Takeuti, Takeshi Tsukada, Soichiro Fujii, Mitsuhiko Fujio, and Hiroyuki Miyoshi for valuable discussion about partial semigroups.

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Authors and Affiliations

  1. Department of Information Systems Creation, Faculty of Engineering, Kanagawa University, Yokohama, Japan

    Koki Nishizawa

  2. Field of Information Systems Creation, Course of Engineering, Graduate School of Engineering, Kanagawa University, Yokohama, Japan

    Koji Yasuda

  3. Department of Mathematics and Computer Science, Kagoshima University, Kagoshima, Japan

    Hitoshi Furusawa

Authors
  1. Koki Nishizawa
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  2. Koji Yasuda
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  3. Hitoshi Furusawa
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Corresponding author

Correspondence to Koki Nishizawa .

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Editors and Affiliations

  1. Laboratoire d’informatique (LIX), École Polytechnique, Palaiseau, France

    Uli Fahrenberg

  2. Faculty of Mathematics, Chapman University, Orange, CA, USA

    Peter Jipsen

  3. Department of Computer Science, Brock University, St. Catharines, ON, Canada

    Michael Winter

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Nishizawa, K., Yasuda, K., Furusawa, H. (2020). Preorders, Partial Semigroups, and Quantales. In: Fahrenberg, U., Jipsen, P., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2020. Lecture Notes in Computer Science(), vol 12062. Springer, Cham. https://doi.org/10.1007/978-3-030-43520-2_15

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  • DOI: https://doi.org/10.1007/978-3-030-43520-2_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43519-6

  • Online ISBN: 978-3-030-43520-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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