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Article | Open Access

Direct Approaches for Representations of Various Algebraic Domains via Closure Spaces

Guojun Wu1(), Luoshan Xu2
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
College of Mathematical Science, Yangzhou University, Yangzhou 225002, China
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Abstract

In this paper, F-augmented closure spaces are generalized to F-closure spaces, and the concept of F-closed sets are introduced. Properties of their ordered structures are investigated. Representations of various algebraic domains such as algebraic lattices, algebraic L-domains, BF-domains via F-closure spaces are considered. As applications of these methods, more direct approaches to representing various algebraic domains via classical closure space are given, respectively. F-relations between F-closure spaces are defined and properties of them are examined. It is also proved that the category of algebraic domains with Scott continuous maps is equivalent to that of F-closure spaces with F-relations.

Keywords

closure spacealgebraic domainalgebraic latticealgebraic L-domainBF-domaincategorical equivalence

References

[1]
D. Scott, Continuous lattices, in Topos, Algebraic Geometry and Logic, F. W. Lawvere Ed. Heidelberg, Germany: Springer, 1972, pp. 97–136.
Crossref
[2]
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains. Cambridge, UK: Cambridge University Press, 2003.
Crossref
[3]

D. Zhao, Closure spaces and completions of posets, Semigroup Forum, vol. 90, no. 2, pp. 545–555, 2015.

Crossref Google Scholar
[4]

Z. Zhang, A general categorical reflection for various completions of Q-closure spaces and Q-ordered sets, Fuzzy Sets Syst., vol. 473, p. 108736, 2023.

Crossref Google Scholar
[5]

Z. Zhang, Q. Li, and N. Zhang, A unified method for completions of posets and closure spaces, Soft Comput., vol. 23, no. 21, pp. 10699–10708, 2019.

Crossref Google Scholar
[6]

Z. Zhang, F. G. Shi, and Q. Li, The R-completion of closure spaces, Topol. Appl., vol. 305, pp. 107873, 2022.

Crossref Google Scholar
[7]

Z. Zou and Q. Li, On subset families that form a continuous lattice, Electron. Notes Theor. Comput. Sci., vol. 333, pp. 163–172, 2017.

Crossref Google Scholar
[8]
G. Birkhoff, Rings of sets, Duke Math. J., vol. 3, no. 3, pp. 443–454, 1937.
Crossref
[9]
M. H. Stone, The theory of representations for Boolean algebras, Trans. Am. Math. Soc., vol. 40, no. 1, pp. 37–111, 1936.
Crossref
[10]
B. A. Davey and H. A. Priestley, Introduction to Lattices and Order. Cambridge, UK: Cambridge University Press, 2002.
Crossref
[11]

Q. Li, L. Wang, and L. Yao, A representation of continuous lattices based on closure spaces, Quaest. Math., vol. 44, no. 11, pp. 1513–1528, 2021.

Crossref Google Scholar
[12]

L. Guo and Q. Li, The categorical equivalence between algebraic domains and F-augmented closure spaces, Order, vol. 32, no. 1, pp. 101–116, 2015.

Crossref Google Scholar
[13]

M. Wu, L. Guo, and Q. Li, New representations of algebraic domains and algebraic L-domains via closure systems, Semigroup Forum, vol. 103, pp. 700–712, 2021.

Crossref Google Scholar
[14]

S. Su and Q. Li, The category of algebraic L-closure systems, J. Intell. Fuzzy Syst., vol. 33, no. 4, pp. 2199–2210, 2017.

Crossref Google Scholar
[15]

L. Yao and Q. Li, Representation of bifinite domains by BF-closure spaces, Math. Slovaca, vol. 71, no. 3, pp. 565–572, 2021.

Crossref Google Scholar
[16]

Q. X. Zhang, Q. G. Li, and L. C. Wang, A representation of arithmetic semilattices by closure spaces, (in Chinese), Fuzzy Syst. Math., vol. 35, no. 2, pp. 1–9, 2021.

Google Scholar
[17]
M. L. J. van de Vel, Theory of Convex Spaces, Amsterdam, the Netherlands: North Holland, 1993.
[18]

C. Shen, S. J. Yang, D. Zhao, and F. G. Shi, Lattice-equivalence of convex spaces, Algebr. Univers., vol. 80, pp. 26, 2019.

Crossref Google Scholar
[19]

W. Yao and C. J. Zhou, Representation of sober convex spaces by join-semilattices, J. Nonlinear Convex Anal., vol. 21, no. 12, pp. 2715–2724, 2020.

Google Scholar
[20]

L. Yao and Q. Li, Representation of FS-domains based on closure spaces, Electron. Notes Theor. Comput. Sci., vol. 345, pp. 271–279, 2019.

Crossref Google Scholar
[21]
L. C. Wang and Q. G. Li, The categorical equivalence between domains and interpolative generalized closure spaces, Stud. Log., vol. 111, pp. 187–215, 2023.
Crossref
[22]

S. W. Wang, B. Zhang, J. Y. Ma, and L. C. Wang, A representation of stably continuous semilattices by closure spaces, (in Chinese), J. Qufu Norm. Univ. (Nat. Sci.), vol. 50, no. 1, pp. 61–66, 2024.

Google Scholar
[23]
A. Jung, Cartesian closed categories of algebraic CPOs, Theor. Comput. Sci., vol. 70, no. 2, pp. 233–250, 1990.
Crossref
[24]
G. L. Jean, Non-Hausdorff Topology and Domain Theory. Cambridge, UK: Cambridge University Press, 2013.
[25]
L. Xu, External characterizations of continuous sL-domains, in Proc. 2nd Int. Symp. Domain Theory, Chengdu, China, 2021, pp. 137–149.
Crossref
[26]
J. Zou, Y. Zhao, C. Miao, and L. Wang, A set-theoretic representation of algebraic L-domains, in Proc. 17th Annu. Conf. Theory and Applications of Models of Computation (TAMC 2022), Tianjin, China, 2022, pp. 370–381.
Crossref
[27]
M. Barr and C. Wells, Category Theory for Computing Science, Montreal, Canada: Les Publications CRM, 1999.
Fuzzy Information and Engineering
Volume 16 Issue 2,
June 2024
Pages 121-143
DOI: 10.26599/FIE.2024.9270036
Cite this article:
Wu G, Xu L. Direct Approaches for Representations of Various Algebraic Domains via Closure Spaces. Fuzzy Information and Engineering, 2024, 16(2): 121-143. https://doi.org/10.26599/FIE.2024.9270036
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