Hostname: page-component-857557d7f7-h6shg Total loading time: 0 Render date: 2025-11-21T22:53:18.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Not every complete lattice can support a unital quantale

Part of: Lattices Ordered sets Ordered structures

Published online by Cambridge University Press:  29 October 2025

Shengwei Han  and
Xin Wang
Shengwei Han*
Affiliation:
Department of Mathematics, Shaanxi Normal University, Xi'an, 710119, China e-mail: wangxin2020@snnu.edu.cn
Xin Wang
Affiliation:
Department of Mathematics, Shaanxi Normal University, Xi'an, 710119, China e-mail: wangxin2020@snnu.edu.cn
*
e-mail: hansw@snnu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Quantales can be regarded as a combination of complete lattices and semigroups. Unital quantales constitute a significant subclass within quantale theory, which play a crucial role in the theoretical framework of quantale research. It is well known that every complete lattice can support a quantale. However, the question of whether every complete lattice can support a unital quantale has not been considered before. In this article, we first give some counter-examples to indicate that the answer to the above question is negative, and then investigate the complete lattices of supporting unital quantales.

Keywords

Complete latticesemigroupquantaleunital quantale

MSC classification

Primary: 06F07: Quantales 06A06: Partial order, general 06B05: Structure theory

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 12
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12471436 and 12331016).

References

Abramsky, S. and Vickers, S., Quantale, observational logic and process semantics . Math. Struct. Comput. Sci. 3(1993), no. 2, 161–227.Google Scholar
Birkhoff, G., Lattice theory. American Mathematical Society, Providence, RI, 1940.Google Scholar
Borceux, F., Rosický, J., and Van Bossche, G., Quantales and ${C}^{\ast }$ -algebras . J. Lond. Math. Soc. 40(1989), no. 3, 398–404.Google Scholar
Connes, A., Noncommutative geometry. Academic Press, San Diego, CA, 1994.Google Scholar
Eklund, P., Gutiérrez García, J., Höhle, U., and Kortelainen, J., Semigroups in complete lattices: Quantales, modules and related topics, Developments in Mathematics, 54, Springer, Berlin, 2018.Google Scholar
Fan, L., A new approach to quantitative domain theory . Electron. Notes Theor. Comput. Sci. 45(2001), 77–87.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M., and Scott, D. S., A compendium of continuous lattices. Springer-Verlag, New York, NY, 1980.Google Scholar
Gutiérrez García, J. and Höhle, U., Unitally nondistributive quantales . Port. Math. 82(2025), nos. 3–4, 259–280.Google Scholar
Han, S. and Lv, J., Construction of unital quantales . Canad. Math. Bull. 68(2025), no. 3, 761–776.Google Scholar
Hofmann, D. and Waszkiewicz, P., Approximation in quantale-enriched categories . Topol. Appl. 158(2011), no. 8, 963–977.Google Scholar
Kruml, D. and Paseka, J., Algebraic and categorical aspects of quantales . Handbook Algebra 5(2008), 323–326.Google Scholar
Kruml, D. and Resende, P., On quantales that classify ${C}^{\ast }$ -algebras . Cah. Topol. Géom. Différ. Catég. 45(2004), no. 4, 287–296.Google Scholar
Mulvey, C. J. and Pelletier, J. W., On the quantisation of points . J. Pure Appl. Algebra 159(2001), nos. 2–3, 231–295.Google Scholar
Mulvey, C. J. and Pelletier, J. W., On the quantisation of spaces . J. Pure Appl. Algebra 175(2002), nos. 1–3, 289–325.Google Scholar
Mulvey, C. J. and Resende, P., A noncommutative theory of Penrose tilings . Internat. J. Theoret. Phys. 44(2005), no. 6, 655–689.Google Scholar
Paseka, J. and Kruml, D., Embeddings of quantales into simple quantales . J. Pure Appl. Algebra 148(2000), no. 2, 209–216.Google Scholar
Protin, M. C. and Resende, P., Quantales of open groupoids . J. Noncommut. Geom. 6(2012), no. 2, 199–247.Google Scholar
Resende, P., Quantales, finite observation and strong bismulation . Theoret. Comput. Sci. 254(2001), nos. 1–2, 95–149.Google Scholar
Resende, P., Étale groupoids and their quantales . Adv. Math. 208(2007), no. 1, 147–209.Google Scholar
Rosenthal, K. I., Quantales and their applications. Longman Scientific & Technical, New York, NY, 1990.Google Scholar
Yetter, D. N., Quantales and (noncommutative) linear logic . J. Symbolic Logic 55(1990), no. 1, 41–64.Google Scholar
Zhang, D., An enriched category approach to many valued topology . Fuzzy Sets and Systems 158(2007), 349–366.Google Scholar