Kybernetika 58 no. 2, 145-162, 2022

Fuzzy sets (in)equations with a complete codomain lattice

Vanja Stepanović and Andreja TepavčevićDOI: 10.14736/kyb-2022-2-0145

Abstract:

The paper applies some properties of the monotonous operators on the complete lattices to problems of the existence and the construction of the solutions to some fuzzy relational equations, inequations, and their systems, taking a complete lattice for the codomain lattice. The existing solutions are extremal - the least or the greatest, thus we prove some extremal problems related to fuzzy sets (in)equations. Also, some properties of upper-continuous lattices are proved and applied to systems of fuzzy sets (in)equations, in a special case of a meet-continuous complete codomain lattice.

Keywords:

fuzzy relations, fuzzy set equations, fuzzy set inequations, monotonous operator, upper continuous lattice

Classification:

03B52, 03E72, 06B23

paper.pdf

References:

  1. B. De Baets: Analytical solution methods for fuzzy relational equations. In: Fundamentals of Fuzzy Sets, in: Handb. Fuzzy Sets Ser. 1 (D. Dubois, H. Prade, eds.), Kluwer Academic Publishers, 2000, pp. 291-340.   CrossRef
  2. P. Cousot and R. Cousot: Constructive Versions of Tarski's Fixed Point Theorem. Pacific J. Math. 82 (1979), 43-57.   DOI:10.2140/pjm.1979.82.43
  3. B. A. Davey and H. A. Priestley: Introduction to Lattices and Order. Cambridge University Press, 1992.   CrossRef
  4. S. Gottwald: Approximate solutions of fuzzy relational equations and a characterization of t-norms that define metrics for fuzzy sets. Fuzzy Sets Syst. 75 (1995), 189-201.   DOI:10.1016/0165-0114(95)00011-9
  5. S. Gottwald: On the existence of solutions of systems of fuzzy equations. Fuzzy Sets Syst. 12 (1984), 301-302.   DOI:10.1016/0165-0114(84)90076-9
  6. S. Gottwald and W. Pedrycz: Solvability of fuzzy relational equations and manipulation of fuzzy data. Fuzzy Sets Syst. 18 (1986), 1-21.   CrossRef
  7. J. Ignjatović, M. Ćirić and S. Bogdanovic: On the greatest solution of weakly linear systems of fuzzy relation inequalities and equations. Fuzzy Sets Syst. 161 (2010), 3081-3113.   DOI:10.1016/j.fss.2010.08.002
  8. J. Ignjatović, M. Ćirić and B. Šešelja: Fuzzy relational inequalities and equalities, fuzzy quasi-orders, closures and openings of fuzzy sets. Fuzzy Sets Syst. 260 (2015), 1-24.   DOI:10.1016/j.fss.2014.05.006
  9. J. Jimenez, S. Montes, B. Šešelja and A. Tepavčević: Fuzzy correspondence inequations and equations. Fuzzy Sets Syst. 239 (2014), 81-90.   DOI:10.1016/j.fss.2012.06.003
  10. J. Jimenez, S. Montes, B. Šešelja and A. Tepavčević: Fuzzy relational inequations and equation in the framework of control problems. In: Symbolic and Quantitative Approaches to Reasoning with Uncertainty (W. Liu, ed.), 2011, pp. 606-615; Lecture Notes in Artificial Intelligence, vol. 6717.   CrossRef
  11. J. Jimenez, S. Montes, B. Šešelja and A. Tepavčević: Lattice valued approach to closed sets under fuzzy relations: theory and application. Comput. Math. Appl. 62 (2011), 3729-3740.   DOI:10.1016/j.camwa.2011.09.021
  12. F. Klawonn: Fuzzy points, fuzzy relations and fuzzy functions. In: Discovering World with Fuzzy Logic (V. Novak, I. Perfilieva, eds.), Physica-Verlag, Heidelberg 2000, pp. 431-453.   CrossRef
  13. I. Perfilieva: Fuzzy function as a solution to a system of fuzzy relation equations. Int. J. Gen. Syst. 32 147 (2003), 361-372.   CrossRef
  14. I. Perfilieva: Fuzzy function as an approximate solution to a system of fuzzy relation equations. Fuzzy Sets Syst. 147 (2004), 363-383.   DOI:10.1016/j.fss.2003.12.007
  15. I. Perfilieva: System of fuzzy relation equations as a continuous model of IF-THEN rules. Inf. Sci. 177 (2007), 3218-3227.   DOI:10.1016/j.ins.2006.11.006
  16. E. Sanchez: Resolution of eigen fuzzy sets equations. Fuzzy Sets Syst. 1 (1978), 69-74.   DOI:10.1016/0165-0114(78)90033-7
  17. E. Sanchez: Solution of fuzzy equations with extended operations. Fuzzy Sets Syst. 12 (1984), 237-248.   DOI:10.1016/0165-0114(84)90071-X
  18. B. Šešelja and A. Tepavčević: Weak Congruences in Universal Algebra. Institute of Mathematics, Novi Sad 2001.   CrossRef
  19. V. Stepanović: Fuzzy set inequations and equations with a meet-continuous codomain lattice. J. Intell. Fuzzy Syst. 34 (2018), 4009-4021.   DOI:10.3233/JIFS-171098
  20. A. Tarski: A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math. 5 (1955), 285-309.   DOI:10.2140/pjm.1955.5.285
  21. A. Tepavčević: Diagonal relation as a continuous element in a weak congruence lattice. In: Proc. International Conference on General Algebra and Ordered Sets, Olomouc 1994, pp. 156-163.   CrossRef