Kybernetika 57 no. 4, 714-736, 2021

Generating methods for principal topologies on bounded lattices

Funda Karaçal, Ümit Ertuğrul and M. Nesibe KesicioğluDOI: 10.14736/kyb-2021-4-0714

Abstract:

In this paper, some generating methods for principal topology are introduced by means of some logical operators such as uninorms and triangular norms and their properties are investigated. Defining a pre-order obtained from the closure operator, the properties of the pre-order are studied.

Keywords:

uninorm, triangular norm, bounded lattice, principal topology, generating method

Classification:

03E72, 03B52, 08A72, 54A10, 06B30, 06F30

References:

  1. P. Alexandroff: Diskrete Räume. Mat. Sb. (N.S.) 2 (1937), 501-518.   CrossRef
  2. G. Birkhoff: Lattice Theory. American Mathematical Society, New York 1948.   CrossRef
  3. X. Chen: Cores of Alexandroff spaces.    CrossRef
  4. G. D. Çaylı, Ü. Ertuğrul, T. Köroğlu and F. Karaçal: Notes on locally internal uninorm on bounded lattices. Kybernetika 53 (2017), 911-921.   DOI:10.14736/kyb-2017-5-0911
  5. I. Dahane, S. Lazaar, T. Richmond and T. Turki: On resolvable primal spaces. Quaestiones Mathematicae (2018), 15-35.   DOI:10.2989/16073606.2018.1437093
  6. J. Dixmier: General Topology. Springer-Verlag, 1984.   CrossRef
  7. O. Echi: Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces. Boll Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8 (2003), 489-507.   CrossRef
  8. O. Echi: The category of flows of Set and Top. Topol. Appl. 159 (2012), 2357-2366.   DOI:10.1016/j.topol.2011.11.059
  9. Ü. Ertuğrul, M. N. Kesicioğlu and F. Karaçal: Ordering based on uninorms. Inform. Sci. 330 (2016), 315-327.   DOI:10.1016/j.ins.2015.10.019
  10. Ü. Ertuğrul, F. Karaçal and R. Mesiar: Modified ordinal sums of triangular norms and triangular conorms on bounded lattices. Int. J. Intell. Systems 30 (2015), 807-817.   DOI:10.1002/int.21713
  11. J. Fodor, R. Yager and A. Rybalov: Structure of uninorms. Int. J. Uncertain. Fuzziness Knowledge-Based Systems 5 (1997), 411-427.   CrossRef
  12. M. Grabisch, J.-L. Marichal, R. Mesiar and E. Pap: Aggregation Functions. Cambridge University Press, 2009.   CrossRef
  13. F. Karaçal and T. Köroğlu: An Alexandroff Topology Obtained from Uninorms. Submitted (2019).   CrossRef
  14. F. Karaçal and R. Mesiar: Uninorms on bounded lattices. Fuzzy Sets and Systems 261 (2015), 33-43.   DOI:10.1016/j.fss.2014.05.001
  15. A. Katsevich and P. Mikusiński: Order of spaces of pseudoquotients. Top. Proc. 44 (2014), 21-31.   CrossRef
  16. J. L. Kelley: General Topology. Springer, New York 1975.   CrossRef
  17. M. N. Kesicioğlu, Ü. Ertuğrul and F. Karaçal: Some notes on U-partial order. Kybernetika 55 (2019), 3, 518-530.   DOI:10.14736/kyb-2019-3-0518
  18. H. Lai and D. Zhang: Fuzzy preorder and fuzzy topology. Fuzzy Sets Systems 157 (2006), 14, 1865-1885.   DOI:10.1016/j.fss.2006.02.013
  19. S. Lazaar, T. Richmond and S. Houssem: The autohomeomorphism group of connected homogeneous functionally Alexandroff spaces. Comm. Algebra (2019).   DOI:10.1080/00927872.2019.1570240
  20. S. Lazaar, T. Richmond and T. Turki: Maps generating the same primal space. Quaest. Math. 40 (2017), 17-28.   DOI:10.2989/16073606.2016.1260067
  21. Z. Ma and W. M. Wu: Logical operators on complete lattices. Information Sciences 55 (1991), 77-97.   DOI:10.1016/0020-0255(91)90007-H
  22. A. K. Steiner: The lattice of topologies: Structure and complementation. Trans. Amer. Math. Soc. 122 (1969), 379-398.   CrossRef
  23. P. Walden: Effective topology from spacetime tomography. J. Physics: Conference Series 68 (2007), 12-28.   CrossRef
  24. R. R. Yager and A. Rybalov: Uninorm aggregation operators. Fuzzy Sets Systems 80 (1996), 111-120.   DOI:10.1016/0165-0114(95)00133-6
  25. H.-P. Zhang, R. Pérez.-Fernández and B. De Baets: Topologies induced by the representatiton of a betweenness relation as a family of order relations. Topol. Appl. 258 (2019), 100-114.   DOI:10.1016/j.topol.2019.02.045