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Soft Computing

, Volume 22, Issue 22, pp 7539–7551 | Cite as

Stratified L-prefilter convergence structures in stratified L-topological spaces

  • Bin Pang
  • Zhen-Yu Xiu
Foundations

Abstract

In this paper, a new approach to fuzzy convergence theory in the framework of stratified L-topological spaces is provided. Firstly, the concept of stratified L-prefilter convergence structures is introduced and it is shown that the resulting category is a Cartesian closed topological category. Secondly, the relations between the category of stratified L-prefilter convergence spaces and the category of stratified L-topological spaces are studied and it is proved that the latter can be embedded in the former as a reflective subcategory. Finally, the relations between the category of stratified L-prefilter convergence spaces and the category of stratified L-Min convergence spaces (fuzzy convergence spaces in the sense of Min) are investigated and it is shown that the former can be embedded in the latter as a reflective subcategory.

Keywords

Fuzzy topology Fuzzy convergence L-prefilter Quasi-coincident neighborhood system Cartesian-closedness Reflective subcategory 

Notes

Acknowledgements

The authors would like to express their sincere thanks to the anonymous reviewers and the area editor for their careful reading and constructive comments. The first author thanks to the Natural Science Foundation of China (No. 11701122) and the Natural Science Foundation of Guangdong Province (No. 2017A030310584). The second author thanks to the Scientific Research Foundation of CUIT (KYTZ201631, CRF201611, 17ZB0093).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of this paper.

References

  1. Adámek J, Herrlich H, Strecker GE (1990) Abstract and concrete categories. Wiley, New YorkzbMATHGoogle Scholar
  2. Chang CL (1968) Fuzzy topological spaces. J Math Anal Appl 24:182–190MathSciNetCrossRefGoogle Scholar
  3. Fang JM (2010a) Stratified \(L\)-ordered convergence structures. Fuzzy Sets Syst 161:2130–2149MathSciNetCrossRefGoogle Scholar
  4. Fang JM (2010b) Relationships between \(L\)-ordered convergence structures and strong \(L\)-topologies. Fuzzy Sets Syst 161:2923–2944MathSciNetCrossRefGoogle Scholar
  5. Güloǧlu M, Coker D (2005) Convergence in \(I\)-fuzzy topological spaces. Fuzzy Sets Syst 151:615–623MathSciNetCrossRefGoogle Scholar
  6. Höhle U, Šostak AP (1999) Axiomatic foundations of fixed-basis fuzzy topology. In: Höhle U, Rodabaugh SE (eds) Mathematics of fuzzy sets: logic, topology, and measure theory, Handbook Series, vol 3. Kluwer, Boston, pp 123–173CrossRefGoogle Scholar
  7. Jäger G (2001) A category of \(L\)-fuzzy convergence spaces. Quaest Math 24:501–517MathSciNetCrossRefGoogle Scholar
  8. Jäger G (2005) Subcategories of lattice-valued convergence spaces. Fuzzy Sets Syst 156:1–24MathSciNetCrossRefGoogle Scholar
  9. Jäger G (2008) Lattice-valued convergence spaces and regularity. Fuzzy Sets Syst 159:2488–2502MathSciNetCrossRefGoogle Scholar
  10. Jäger G (2010a) Compactification of lattice-valued convergence spaces. Fuzzy Sets Syst 161:1002–1010MathSciNetCrossRefGoogle Scholar
  11. Jäger G (2010b) Compactness in lattice-valued function spaces. Fuzzy Sets Syst 161:2962–2974MathSciNetCrossRefGoogle Scholar
  12. Jäger G (2012a) A one-point compactification for lattice-valued convergence spaces. Fuzzy Sets Syst 190:21–31MathSciNetCrossRefGoogle Scholar
  13. Jäger G (2012b) Largest and smallest T\(_2\)-compactifications of lattice-valued convergence spaces. Fuzzy Sets Syst 190:32–46CrossRefGoogle Scholar
  14. Jäger G (2016) Connectedness and local connectedness for lattice-valued convergence spaces. Fuzzy Sets Syst 300:134–146MathSciNetCrossRefGoogle Scholar
  15. Li LQ, Jin Q (2011) On adjunctions between Lim, S\(L\)-Top, and S\(L\)-Lim. Fuzzy Sets Syst 182:66–78MathSciNetCrossRefGoogle Scholar
  16. Li LQ, Jin Q (2012) On stratified \(L\)-convergence spaces: pretopological axioms and diagonal axioms. Fuzzy Sets Syst 204:40–52MathSciNetCrossRefGoogle Scholar
  17. Li LQ, Jin Q (2014) \(p\)-Topologicalness and \(p\)-regularity for lattice-valued convergence spaces. Fuzzy Sets Syst 238:26–45MathSciNetCrossRefGoogle Scholar
  18. Li LQ, Jin Q, Hu K (2015) On stratified \(L\)-convergence spaces: Fischer’s diagonal axiom. Fuzzy Sets Syst 267:31–40MathSciNetCrossRefGoogle Scholar
  19. Li LQ, Jin Q, Meng GW, Hu K (2016) The lower and upper \(p\)-topological (\(p\)-regular) modifications for lattice-valued convergence spaces. Fuzzy Sets Syst 282:47–51MathSciNetCrossRefGoogle Scholar
  20. Lowen R (1979) Convergence in fuzzy topological spaces. Gen Topl Appl 10:147–160MathSciNetCrossRefGoogle Scholar
  21. Lowen E, Lowen R, Wuyts P (1991) The categorical topological approach to fuzzy topology and fuzzy convergence. Fuzzy Sets Syst 40:347–373CrossRefGoogle Scholar
  22. Lowen E, Lowen R (1992) A topological universe extension of FTS. In: Rodabaugh SE, Klement EP, Höhle U (eds) Applications of category theory to fuzzy sets. Kluwer, DordrechtzbMATHGoogle Scholar
  23. Min KC (1989) Fuzzy limit spaces. Fuzzy Sets Syst 32:343–357MathSciNetCrossRefGoogle Scholar
  24. Pang B, Fang JM (2011) \(L\)-fuzzy Q-convergence structures. Fuzzy Sets Syst 182:53–65MathSciNetCrossRefGoogle Scholar
  25. Pang B (2013) Further study on \(L\)-fuzzy Q-convergence structures. Iran J Fuzzy Syst 10(5):147–164MathSciNetzbMATHGoogle Scholar
  26. Pang B (2014) On \((L, M)\)-fuzzy convergence spaces. Fuzzy Sets Syst 238:46–70CrossRefGoogle Scholar
  27. Pang B, Shi F-G (2014) Degrees of compactness of \((L, M)\)-fuzzy convergence spaces and its applications. Fuzzy Sets Syst 251:1–22MathSciNetCrossRefGoogle Scholar
  28. Pang B (2014) Enriched \((L, M)\)-fuzzy convergence spaces. J Intell Fuzzy Syst 27:93–103MathSciNetzbMATHGoogle Scholar
  29. Pang B, Zhao Y (2016) Stratified \((L, M)\)-fuzzy Q-convergence spaces. Iran J Fuzzy Syst 14(4):95–111MathSciNetzbMATHGoogle Scholar
  30. Pang B, Zhao Y (2017) Several types of enriched \((L, M)\)-fuzzy convergence spaces. Fuzzy Sets Syst 321:55–72MathSciNetCrossRefGoogle Scholar
  31. Pu BM, Liu YM (1980) Fuzzy topology (I), neighborhood structure of a fuzzy point and Moore–Smith convergence. J Math Anal Appl 76:571–599MathSciNetCrossRefGoogle Scholar
  32. Preuss G (2002) Foundations of topology: an approach to convenient topology. Kluwer, DordrechtCrossRefGoogle Scholar
  33. Wu WC, Fang JM (2012) \(L\)-ordered fuzzifying convergence spaces. Iran J Fuzzy Syst 9(2):147–161MathSciNetzbMATHGoogle Scholar
  34. Xu LS (2001) Characterizations of fuzzifying topologies by some limit structures. Fuzzy Sets Syst 123:169–176MathSciNetCrossRefGoogle Scholar
  35. Yao W (2008) On many-valued stratified \(L\)-fuzzy convergence spaces. Fuzzy Sets Syst 159:2503–2519MathSciNetCrossRefGoogle Scholar
  36. Yao W (2009) On \(L\)-fuzzifying convergence spaces. Iran J Fuzzy Syst 6(1):63–80MathSciNetzbMATHGoogle Scholar
  37. Yao W (2012) Moore–Smith convergence in \((L, M)\)-fuzzy topology. Fuzzy Sets Syst 190:47–62MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingPeople’s Republic of China
  2. 2.College of Applied MathematicsChengdu University of Information TechnologyChengduPeople’s Republic of China

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