Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-04-24T16:42:27.289Z Has data issue: false hasContentIssue false
  • English
  • Français

sL-approximation spaces capture sL-domains

Published online by Cambridge University Press:  04 April 2025

Guojun Wu Open the ORCID record for Guojun Wu [Opens in a new window] ,
Luoshan Xu  and
Wei Yao
Guojun Wu*
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China
Luoshan Xu
Affiliation:
College of Mathematical Science, Yangzhou University, Yangzhou, China
Wei Yao
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China
*
Corresponding author: Guojun Wu; Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, by means of upper approximation operators in rough set theory, we study representations for sL-domains and its special subclasses. We introduce the concepts of sL-approximation spaces, L-approximation spaces, and bc-approximation spaces, which are special types of CF-approximation spaces. We prove that the collection of CF-closed sets in an sL-approximation space (resp., an L-approximation space, a bc-approximation space) ordered by set-theoretic inclusion is an sL-domain (resp., an L-domain, a bc-domain); conversely, every sL-domain (resp., L-domain, bc-domain) is order-isomorphic to the collection of CF-closed sets of an sL-approximation space (resp., an L-approximation space, a bc-approximation space). Consequently, we establish an equivalence between the category of sL-domains (resp., L-domains) with Scott continuous mappings and that of sL-approximation spaces (resp., L-approximation spaces) with CF-approximable relations.

Keywords

Upper approximation operatorsL-domainsL-approximation spaceL-domaincategorical equivalence
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 35 , 2025 , e4
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

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

*

Supported by the National Natural Science Foundation of China (12231007, 12371462, 11671008), the Natural Science Foundation of Jiangsu Province (BK20170483).

References

Barr, M. and Wells, C. (1990). Category Theory for Computing Science, 3rd ed., Englewood Cliffs, Prentice Hall.Google Scholar
Davey, B. A. and Priestley, H. A. (2002). Introduction to Lattices and Order, Cambridge, Cambridge University Press.Google Scholar
Eklund, P., Galán, M. A. and Gähler, W. (2009). Partially ordered monads for monadic topologies, rough sets and Kleene algebras, Electronic Notes in Theoretical Computer Science 225 67–81.CrossRefGoogle Scholar
Ganter, B. and Wille, R. (1999). Formal Concept Analysis, Berlin, Springer-Verlag.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003). Continuous Lattices and Domains, Cambridge, Cambridge University Press.Google Scholar
Goubault-Larrecq, J. (2013). Non-Hausdorff Topology and Domain Theory, Cambridge, Cambridge University Press.CrossRefGoogle Scholar
He, Q. and Xu, L. (2019). Weak algebraic information systems and a new equivalent category of DOM of domains, Theoretical Computer Science 763 1–19.Google Scholar
Järvinen, J. (2007). Lattice theory for rough sets. In: Peters, J. F. et al. (ed.) Transactions on Rough Sets VI, Lecture Notes in Computer Science, Berlin, Springer-Verlag, 400–498.CrossRefGoogle Scholar
Jung, A. (1989). Cartesian closed categories of domains. In: CWI Tracts, vol. 66, Amesterdam, Centrum voor Wiskunde en Informatica.Google Scholar
Jung, A. (1990a). The clssification of continuous domains. In: Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science, Philadelphia, IEEE Computer Society Press, 35–40.Google Scholar
Jung, A. (1990b). Cartesian closed categories of algebraic cpos. Theoretical Computer Science 70 (2) 233–250.CrossRefGoogle Scholar
Khan, M. A. and Patel, V. S. (2022). A formal study of a generalized rough set model based on subset approximation structure. International Journal of Approximate Reasoning 140 52–74.Google Scholar
Kondo, M. (2006). On the structure of generalized rough sets. Information Sciences 176 (5) 589–600.CrossRefGoogle Scholar
Lawson, J. and Xu, L. (2003). When does the class $[\mathcal {A}\to \mathcal {B}]$ consist of continuous domains. Topology and its Applications 130 91–97.CrossRefGoogle Scholar
Lei, Y. and Luo, M. (2009). Rough concept lattices and domains. Annals of Pure and Applied Logic 159 (3) 333–340.CrossRefGoogle Scholar
Liu, G. and Zhu, W. (2008). The algebraic structures of generalized rough set theory. Information Sciences 178 (21) 4105–4113.CrossRefGoogle Scholar
Mao, X. and Xu, L. (2005). Representation theorems for directed completions of consistent algebraic L-domains. Algebra Universalis 54 (4) 435–447.Google Scholar
Pawlak, Z. (1982). Rough sets. International Journal of Foundations of Computer Science 11 (5) 341–356.CrossRefGoogle Scholar
Rong, Y. and Xu, L. (2017). Join reductions and join saturation reductions of abstract knowledge bases. International Journal of Contemporary Mathematical Sciences 12 109–115.CrossRefGoogle Scholar
Scott, D. S. (1972). Continuous lattices. In: Topos, Algebraic Geometry and Logic, Lecture Notes in Mathematics, vol. 274, 97–136.Google Scholar
Spreen, D. (2021). Generalised information systems capture L-domains. Theoretical Computer Science 869 1–28.CrossRefGoogle Scholar
Spreen, D., Xu, L. and Mao, X. (2008). Information systems revisited–the general continuous case, Theoretical Computer Science 405 176–187.CrossRefGoogle Scholar
Wang, L. and Li, Q. (2020). A representation of proper bc-domains based on conjunctive sequent calculi. Mathematical Structures in Computer Science 30 (1) 1–13.Google Scholar
Wang, L. and Li, Q. (2024). L-domains as locally continuous sequent calculi. Archive For Mathematical Logic 63 (3-4) 405–425.CrossRefGoogle Scholar
Wang, L., Li, Q. and Zhou, X. (2021). Continuous L-domains in logical form. Annals of Pure and Applied Logic 172 (9) 1–24.CrossRefGoogle Scholar
Wang, L., Zhou, X. and Li, Q. (2022). Information systems for continuous semi-lattices, Theoretical Computer Science 913 138–150.CrossRefGoogle Scholar
Wang, L., Zhou, X. and Li, Q. (2023). Topological representations of Lawson compact algebraic L-domains and Scott domains. Algebra Universalis 84 (3) 23.Google Scholar
Wang, S. and Li, Q. (2022). A representation of L-domain by formal concept analysis. Soft Computing 26 (19) 9751–9760.Google Scholar
Wu, G. and Xu, L. (2023). Representations of domains via CF-approximation spaces. In: Electronic Notes in Theoretical Informatics and Computer Science, Vol 2-Proceedings of ISDT’9, 13.Google Scholar
Wu, G. and Xu, L. (2024). Direct approaches for representations of various algebraic domains via closure spaces. Fuzzy Information and Engineering 16 (2) 121–143.Google Scholar
Wu, M., Guo, L. and Li, Q. (2016). A representation of L-domains by information systems. Theoretical Computer Science 612 126–136.Google Scholar
Xu, L. (2003) External characterizations of continuous sL-domains. In: Zhang, G., Lawson, J., Liu, Y. and Luo, M. (eds.) Domain Theory, Logic and Computation, Semantic Structures in Computation, vol. 3. Dordrecht, Springer, 137–149.Google Scholar
Xu, L. and Mao, X. (2008a). Formal topological characterizations of various continuous domains. Computers & Mathematics with Applications 56 (2) 444–452.Google Scholar
Xu, L. and Mao, X. (2008b). When do abstract bases generate continuous lattices and L-domains. Algebra Universalis 58 (1) 95–104.Google Scholar
Yang, L. and Xu, L. (2009). Algebraic aspects of generalized approximation spaces. International Journal of Approximate Reasoning 51 (1) 151–161.Google Scholar
Yang, L. and Xu, L. (2011). Topological properties of generalized approximation spaces. Information Sciences 181, 3570–3580.Google Scholar
Zhou, X., Wang, L. and Li, Q. (2024). A direct approach to representing algebraic domains by formal contexts. International Journal of Approximate Reasoning 164 109085.Google Scholar
Zou, J., Zhao, Y., Miao, C. and Wang, L. (2024). A set-theoretic approach to algebraic L-domains. Mathematical Structures in Computer Science 34 (3) 244–257.Google Scholar
Zou, Z., Li, Q. and Ho, W. K. (2018). Domains via approximation operators. Logical Methods In Computer Science 14, 1–17.Google Scholar