Skip to main content
Top
Published in: Soft Computing 21/2019

23-04-2019 | Foundations

A unified method for completions of posets and closure spaces

Authors: Zhongxi Zhang, Qingguo Li, Nan Zhang

Published in: Soft Computing | Issue 21/2019

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we introduce the notion of a subset system on the category CL\(_{0}\) of all \(T_{0}\) closure spaces. And for each subset system Z, the concept of a Z-convergence space arises as a generalization of sober spaces and monotone convergence spaces. We define a Z-completion of a \(T_{0}\) closure space X to be a Z-convergence space \(X_{Z}\) together with a continuous function from X to \(X_{Z}\) satisfying the universal property. In the case that Z is a hereditary subset system, we prove that: (1) for each \(T_{0}\) closure space X, the set of all Z-tapered closed subsets of X endowed with the corresponding closure system is a Z-completion of X; (2) the category CS\(_{Z}\) of all Z-convergence spaces is reflective in the category CL\(_{0}\). The Dedekind–MacNeille completion, the Alexandroff completion, the Frink ideal completion, the ideal completion, the Z-completion for posets, the D-completion, the sobrification, the directed completion, etc., are special cases of the Z-completion.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
go back to reference Gierz G, Hofmann KH, Keimel K, Lawson JD, Mislove M, Scott DS (2003) Continuous lattices and domains. Encyclopedia of mathematics and its applications, vol 93. Cambridge University Press, CambridgeCrossRef Gierz G, Hofmann KH, Keimel K, Lawson JD, Mislove M, Scott DS (2003) Continuous lattices and domains. Encyclopedia of mathematics and its applications, vol 93. Cambridge University Press, CambridgeCrossRef
go back to reference Huang M, Li Q, Li J (2009) Generalized continuous posets and a new cartesian closed category. Appl Categ Struct 17(1):29–42MathSciNetCrossRef Huang M, Li Q, Li J (2009) Generalized continuous posets and a new cartesian closed category. Appl Categ Struct 17(1):29–42MathSciNetCrossRef
go back to reference Johnstone PT (1981) Scott is not always sober. Lect Notes Math 871:282–283CrossRef Johnstone PT (1981) Scott is not always sober. Lect Notes Math 871:282–283CrossRef
go back to reference Wright JB, Wagner EG, Thatcher JW (1978) A uniform approach to inductive posets and inductive closure. Theor Comput Sci 7(1):57–77MathSciNetCrossRef Wright JB, Wagner EG, Thatcher JW (1978) A uniform approach to inductive posets and inductive closure. Theor Comput Sci 7(1):57–77MathSciNetCrossRef
go back to reference Wyler O (1981) Dedekind complete posets and Scott topologies. In: Banaschewsk B, Hoffmann R-E (eds) Continuous lattices proceedings, Bremen 1979, vol 871. Springer, Berlin, pp 384–389 Wyler O (1981) Dedekind complete posets and Scott topologies. In: Banaschewsk B, Hoffmann R-E (eds) Continuous lattices proceedings, Bremen 1979, vol 871. Springer, Berlin, pp 384–389
go back to reference Xu L (2000) Consistently continuous posets and their directed completions. J Yangzhou Univ (Nat Sci Ed) 3(1):1–6 (in Chinese)MathSciNet Xu L (2000) Consistently continuous posets and their directed completions. J Yangzhou Univ (Nat Sci Ed) 3(1):1–6 (in Chinese)MathSciNet
go back to reference Zhang Z, Li Q (2017a) A direct characterization of the monotone convergence space completion. Topol Appl 230:99–104MathSciNetCrossRef Zhang Z, Li Q (2017a) A direct characterization of the monotone convergence space completion. Topol Appl 230:99–104MathSciNetCrossRef
Metadata
Title
A unified method for completions of posets and closure spaces
Authors
Zhongxi Zhang
Qingguo Li
Nan Zhang
Publication date
23-04-2019
Publisher
Springer Berlin Heidelberg
Published in
Soft Computing / Issue 21/2019
Print ISSN: 1432-7643
Electronic ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-019-04009-z

Other articles of this Issue 21/2019

Soft Computing 21/2019 Go to the issue

Premium Partner