Fuzzy Ideals and Fuzzy Filters on Topologies Generated by Fuzzy Relations

Main Article Content

Kheir Saadaoui
Soheyb Milles
Lemnaouar Zedam

Abstract

Recently, Mishra and Srivastava have introduced and studied the notion of fuzzy topology generated by fuzzy relation and several properties were proved. In this paper, we mainly investigate the lattice structure of fuzzy open sets in this topology, and show its various properties and characteristics. Additionally, we extend to this lattice the notions of fuzzy ideal and fuzzy filter. For each of these notions, we fully characterize them in terms of this lattice meet and join operations.

Article Details

References

  1. A. Bennoui, L. Zedam, S. Milles, Several Types of Single-Valued Neutrosophic Ideals and Filters on a Lattice, TWMS J. App. Eng. Math. In Press.
  2. S. Boudaoud, L. Zedam, S. Milles, Principal Intuitionistic Fuzzy Ideals and Filters on a Lattice, Discuss. Math.: Gen. Algebra Appl. 40 (2020), 75–88. https://doi.org/10.7151/dmgaa.1325.
  3. N. Bourbaki, Topologie générale, Springer-Verlag, Berlin Heidelberg, 2007.
  4. C.L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl. 24 (1968), 182–190. https://doi.org/10.1016/0022-247x(68)90057-7.
  5. B. Davvaz, O. Kazanci, A New Kind of Fuzzy Sublattice (Ideal, Filter) of a Lattice, Int. J. Fuzzy Syst. 13 (2011), 55–63.
  6. J.A. Goguen, The Logic of Inexact Concepts, Synthese, 19 (1969), 325–373. https://www.jstor.org/stable/20114646.
  7. B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Second Edition, Cambridge University Press, Cambridge, 2002.
  8. J.L. Kelley, General Topology, Van Nostrand, Princeton, New Jersey, 1955.
  9. Y. Liu, M. Zheng, Characterizations of Fuzzy Ideals in Coresiduated Lattices, Adv. Math. Phys. 2016 (2016), 6423735. https://doi.org/10.1155/2016/6423735.
  10. S. Milles, E. Rak, L. Zedam, Intuitionistic Fuzzy Complete Lattices, in: K.T. Atanassov, O. Castillo, J. Kacprzyk, M. Krawczak, P. Melin, S. Sotirov, E. Sotirova, E. Szmidt, G. De Tré, S. Zadrożny (Eds.), Novel Developments in Uncertainty Representation and Processing, Springer International Publishing, Cham, 2016: pp. 149–160. https://doi.org/10.1007/978-3-319-26211-6_13.
  11. S. Mishra, R. Srivastava, Fuzzy Topologies Generated by Fuzzy Relations, Soft Comput. 22 (2016), 373–385. https://doi.org/10.1007/s00500-016-2458-6.
  12. S. Milles, L. Zedam, E. Rak, Characterizations of Intuitionistic Fuzzy Ideals and Filters Based on Lattice Operations, J. Fuzzy Set Valued Anal. 2017 (2017), 143–159. https://doi.org/10.5899/2017/jfsva-00399.
  13. B.S. Schröder, Ordered Sets, Birkhauser, Boston, USA, 2002.
  14. M.H. Stone, The Theory of Representations of Boolean Algebras, Trans. Amer. Math. Soc. 40 (1936), 37–111. https://doi.org/10.2307/1989664.
  15. M. Tonga, Maximality on Fuzzy Filters of Lattices, Afr. Mat. 22 (2011), 105–114. https://doi.org/10.1007/s13370-011-0009-y.
  16. B. Van Gasse, G. Deschrijver, C. Cornelis, E.E. Kerre, Filters of Residuated Lattices and Triangle Algebras, Inform. Sci. 180 (2010), 3006–3020. https://doi.org/10.1016/j.ins.2010.04.010.
  17. S. Willard, General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts, 1970.
  18. L.A. Zadeh, Fuzzy Sets, Inform. Control. 8 (1965), 331–352.
  19. L.A. Zadeh, Similarity Relations and Fuzzy Orderings, Inform. Sci. 3 (1971), 177–200. https://doi.org/10.1016/s0020-0255(71)80005-1.
  20. L. Zedam, S. Milles, A. Bennoui, Ideals and Filters on a Lattice in Neutrosophic Setting, Appl. Appl. Math. 16 (2021), 1140–1154.