Hostname: page-component-f554764f5-fnl2l Total loading time: 0 Render date: 2025-04-20T14:20:06.188Z Has data issue: false hasContentIssue false
  • English
  • Français

A set-theoretic approach to algebraic L-domains

Published online by Cambridge University Press:  11 April 2024

Juan Zou ,
Yuhan Zhao ,
Cuixia Miao  and
Longchun Wang Open the ORCID record for Longchun Wang [Opens in a new window]
Juan Zou
Affiliation:
School of Mathematical Sciences, Qufu Normal University, Qufu, China
Yuhan Zhao
Affiliation:
School of Mathematical Sciences, Qufu Normal University, Qufu, China
Cuixia Miao
Affiliation:
School of Mathematical Sciences, Qufu Normal University, Qufu, China
Longchun Wang*
Affiliation:
School of Mathematical Sciences, Qufu Normal University, Qufu, China
*
Corresponding author: Longchun Wang; Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, the notion of locally algebraic intersection structure is introduced for algebraic L-domains. Essentially, every locally algebraic intersection structure is a family of sets, which forms an algebraic L-domain ordered by inclusion. It is shown that there is a locally algebraic intersection structure which is order-isomorphic to a given algebraic L-domain. This result extends the classic Stone’s representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it can be seen that many well-known representations of algebraic L-domains, such as logical algebras, information systems, closure spaces, and formal concept analysis, can be analyzed in the framework of locally algebraic intersection structures. Then, a set-theoretic uniformity across different representations of algebraic L-domains is established.

Keywords

Stone’s representation theoremdomain theoryalgebraic L-domainalgebraic intersection structure
Type
Special Issue: TAMC 2022
Information
Mathematical Structures in Computer Science , Volume 34 , Special Issue 3: Theory and Applications of Models of Computation , March 2024 , pp. 244 - 257
Check for updates
Copyright
© The Author(s), 2024. 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 Shandong Provincial Natural Science Foundation (ZR2023MA051, ZR2022MA022).

References

Abramsky, S. (1987). Domain Theory and the Logic of Observable Properties. Phd thesis, University of London.Google Scholar
Abramsky, S. (1991). Domain theory in logical form. Annals of Pure and Applied Logic 51 1–77.CrossRefGoogle Scholar
Birkhoff, G. (1937). Rings of sets. Duke Mathematical Journal 3 (3) 443–454.CrossRefGoogle Scholar
Chen, Y. (1997). Stone duality and representation of stable domain. Computers and Mathematics with Applications 34 (1) 27–41.CrossRefGoogle Scholar
Chen, Y. and Jung, A. (2006). A logical approach to stable domains. Theoretical Computer Science 368 124–148.CrossRefGoogle Scholar
Davey, B. A. and Priestly, H. A. (2002). Introduction to Lattices and Order, Cambridge, Cambridge University Press.CrossRefGoogle Scholar
Ganter, B. and Wille, R. (1999). Formal Concept Analysis, Berlin, Springer-Verlag.CrossRefGoogle 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.CrossRefGoogle Scholar
Guo, L., Li, Q. and Yao, L. (2018). Locally complete consistent F-augmented contexts: A category-theoretic representation of algebraic L-domains. Discrete Applied Mathematics 249 53–63.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.CrossRefGoogle Scholar
Hitzler, P., Kröetzsch, M. and Zhang, G. (2006). A categorical view on algebraic lattices in formal concept analysis. Fundamenta Informaticae 74 1–29.Google Scholar
Huang, M., Li, Q. and Guo, L. (2014). Formal contexts for algebraic domains. Electronic Notes in Theoretical Computer Science 301 79–90.CrossRefGoogle Scholar
Jung, A. (1989). Cartesian Closed Categories of Domains , CWI Tracts, vol. 66, Amesterdam, Centrum voor Wiskunde en Informatica.Google Scholar
Jung, A., Kegelmann, M. and Moshier, M. A. (1999). Multi lingual sequent calculus and coherent spaces. Fundamenta Informaticae 37 369–412.CrossRefGoogle Scholar
Larsen, K. G. and Winskel, G. (1991). Using information systems to solve recursive domain equations effectively. Information and Computation 91 232–258.CrossRefGoogle Scholar
Rounds, W. and Zhang, G. (2001). Clausal logic and logic programming in algebraic domains. Information and Computation 171 183–200.CrossRefGoogle Scholar
Scott, D. S. (1982). Domains for denotational semantics. Lecture Notes in Computer Science 140 577–613.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
Spreen, D. (2021). Generalised information systems capture L-domains. Theoretical Computer Science 869 1–28.CrossRefGoogle Scholar
Stone, M. H. (1936). The theory of representations for Boolean algebras. Transactions of the American Mathematical Society 40 37–111.Google Scholar
Vickers, S. (2004). Entailment systems for stably locally compact locales. Theoretical Computer Science 316 259–296.CrossRefGoogle Scholar
Wang, L. and Li, Q. (2021). Representations of stably continuous semi-lattices by information systems and abstract bases. Information Processing Letters 165 106036.CrossRefGoogle Scholar
Wang, L. and Li, Q. (2022). Bounded complete domains and their logical form. Information and Computation 289 104958.CrossRefGoogle Scholar
Wang, L. and Li, Q. (2023). The categorical equivalence between domains and interpolative generalized closure spaces. Studia Logica 111 187–215.CrossRefGoogle Scholar
Wang, L. and Li, Q. (2024). L-domains as locally continuous sequent calculi. Archive for Mathematical Logic, http://doi.org/10.1007/s00153-023-00903-4.CrossRefGoogle Scholar
Wang, L., Zhou, X. and Li, Q. (2022). Information systems for continuous semi-lattices. Theoretical Computer Science 913 138–150.CrossRefGoogle Scholar
Wu, M., Guo, L. and Li, Q. (2016). A representation of L-domains by information systems. Theoretical Computer Science 612 126–136.CrossRefGoogle Scholar
Wu, M., Guo, L. and Li, Q. (2021). New representations of algebraic domains and algebraic L-domains via closure systems. Semigroup Forum 103 700–712.CrossRefGoogle Scholar
Zhang, G.-Q. (1991). Logic of Domains, Boston, Birkhauser.CrossRefGoogle Scholar