Fuzzy proximal relator spaces

  • Mehmet Ali Öztürk
  • Ebubekir İnan
  • Özlem Tekin
  • James Francis Peters
Original Article
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Abstract

This paper introduces fuzzy Lodato–Smirnov proximal relator spaces. We define fuzzy proximity relation to evaluate the proximity of the sets. Also, it is shown that the max–min composition of fuzzy proximity relations is associative and the Smirnov proximity measure on the Lodato proximity space is a fuzzy spatial Lodato proximity relation. Fuzzy proximity relation approach can be applied to some problems. Therefore, it is used to solve a few issues such as precise sorting of near sets, solution of classification problems and given examples for the problems.

Keywords

Proximity space Relator space Fuzzy relation Fuzzy proximity 

Mathematics Subject Classification

03E72 54E05 90B50 

Notes

Acknowledgements

The authors are thankful for financial support from the Research Fund of Adiyaman University by the Project FEFMAP/2015-0010.

Compliance with ethical standards

Conflict of interest

All authors of this article have no relationships or interests that could have direct or potential influence or impart bias on any aspect of this work.

References

  1. 1.
    Dochviri I, Peters JF (2015) Topological Sorting of finitely many near sets. Math Comput Sci. 1–6, communicatedGoogle Scholar
  2. 2.
    Efremoviĉh VA (1952) The geometry of proximity. Math Sb (N.S) 73(1):189–200 (in Russian)MathSciNetMATHGoogle Scholar
  3. 3.
    Hussain M (2010) Fuzzy Relations. Master Thesis, Department of Mathematics and Science, Supervisor E. R. AnderssonGoogle Scholar
  4. 4.
    İnan E (2017) Approximately semigroups and ideals: an algebraic view of digital images. Afyon Kocatepe University J Sci Eng 17:479–487CrossRefGoogle Scholar
  5. 5.
    Kaufmann A (1975) Introduction to the theory of fuzzy subsets, vol 1. Academic Press, New YorkMATHGoogle Scholar
  6. 6.
    Leader S (1959) On clusters in proximity spaces. Fundam Math 47:205–213MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lodato MW (1962) On topologically induced generalized proximity relations. Ph.D. thesis, Rutgers University, 42pp, supervisor: S. LeaderGoogle Scholar
  8. 8.
    Lodato MW (1964) On topologically induced generalized proximity relations I. Proc Am Math Soc 15:417–422MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lodato MW (1966) On topologically induced generalized proximity relations II. Pacific J Math 17:131–135MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Naimpally SA, Warrack BD (1970) Proximity spaces. Cambridge University, CambridgeMATHGoogle Scholar
  11. 11.
    Peters JF (2016) Computational proximity. Excursions in the topology of digital images. Intelligent Systems Reference Library, Springer, BerlinGoogle Scholar
  12. 12.
    Peters JF (2013) Near sets: an introduction. Math Comp Sci 7:3–9MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Peters JF (2016) Proximal relator spaces. Filomat 30(2):469–472MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Peters JF (2014) Topology of digital images. Visual pattern discovery in proximity spaces. Intelligent Systems Reference Library, vol. 63, Springer, BerlinGoogle Scholar
  15. 15.
    Smirnov JM (1964) On proximity spaces. Math Sb (N.S.) 31(73):543–574. English translation: Am Math Soc Trans Ser 2(38):5–35 (1964)Google Scholar
  16. 16.
    Szás À (1987) Basic tools and mild continuties in relator spaces. Acta Math Hungar 50:177–201MathSciNetCrossRefGoogle Scholar
  17. 17.
    Szás À, Zakaria A (2016) Mild continuity properties of relations and relators in relator spaces. Essays Math Appl 439–511Google Scholar
  18. 18.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefMATHGoogle Scholar
  19. 19.
    Zadeh LA (1971) Similarity relations and fuzzy orderings. Inform Sci 3:177–206MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Zhan J, Liu Q, Davvaz B (2015) A new rough set theory: rough soft hemirings. J Intell Fuzzy Syst 28:1687–1697MathSciNetMATHGoogle Scholar
  21. 21.
    Zhan J, Bin Y, Fotea VE (2016) Characterizations of two kinds of hemirings based on probability spaces. Soft Comput 20:637648CrossRefMATHGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2018

Authors and Affiliations

  • Mehmet Ali Öztürk
    • 1
  • Ebubekir İnan
    • 1
  • Özlem Tekin
    • 1
  • James Francis Peters
    • 2
  1. 1.Department of Mathematics, Faculty of Arts and SciencesAdıyaman UniversityAdıyamanTurkey
  2. 2.Computational Intelligence Laboratory, Department of Electrical and Computer EngineeringUniversity of ManitobaWinnipegCanada

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