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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, London (1971)
He, J., Hoare, C.A.R.: Weakest prespecification. Inf. Process. Lett. 24, 71–76 (1987)
Vickers, S.: Topology via Logic. Cambridge University Press, Cambridge (1989)
Nishizawa, K., Furusawa, H.: Relational representation theorem for powerset quantales. In: Kahl, W., Griffin, T.G. (eds.) RAMICS 2012. LNCS, vol. 7560, pp. 207–218. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33314-9_14
Pratt, V.R.: Dynamic algebras and the nature of induction. In: Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, STOC 1980, pp. 22–28. Association for Computing Machinery, New York (1980)
Mulvey, C.J.: In: Second Topology Conference. Rendiconti del Circolo Matematico di Palermo, vol. 2, no. 12, pp. 99–104 (1986)
Rosenthal, K.: Quantales and Their Applications. Pitman Research Notes in Mathematics, vol. 234. Longman Scientific & Technical (1990)
Eklund, P., Gutie Rrez Garcia, J., Hoehle, U., Kortelainen, J.: Semigroups in Complete Lattices: Quantales, Modules and Related Topics, vol. 54 (2018)
Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994)
Jonsson, D.: Poloids from the points of view of partial transformations and category theory (2017). https://arxiv.org/abs/1710.04634
Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)
Acknowledgements
The authors thank Izumi Takeuti, Takeshi Tsukada, Soichiro Fujii, Mitsuhiko Fujio, and Hiroyuki Miyoshi for valuable discussion about partial semigroups.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-43520-2_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-43519-6
Online ISBN: 978-3-030-43520-2
eBook Packages: Computer ScienceComputer Science (R0)