Kybernetika 51 no. 2, 255-267, 2015

Quotient algebraic structures on the set of fuzzy numbers

Dorina Fechete and Ioan FecheteDOI: 10.14736/kyb-2015-2-0255

Abstract:

A. M. Bica has constructed in \cite{Bica 2007} two isomorphic Abelian groups, defined on quotient sets of the set of those unimodal fuzzy numbers which have strictly monotone and continuous sides. In this paper, we extend the results of above mentioned paper, to a larger class of fuzzy numbers, by adding the flat fuzzy numbers. Furthermore, we add the topological structure and we characterize the constructed quotient groups, by using the set of the continuous functions with bounded variation, defined on $[0,1]$.

Keywords:

fuzzy number, function with bounded variation, semigroup (monoid) with involution, topological group, metric space

Classification:

08A72, 54H11

References:

  1. A. I. Ban and B. Bede: Power series of fuzzy numbers withy cross product and applications to fuzzy differential equations. J. Concrete Appl. Math 4 (2006), 2, 125-152.   CrossRef
  2. A. I. Ban and A. Bica: Solving systems of equivalentions. J. Appl. Math. Comput. 20 (2006), 1-2, 97-118.   DOI:10.1007/bf02831926
  3. B. Bede: Mathematics of Fuzzy Sets and Fuzzy Logic. Springer-Verlag, Berlin 2013.   CrossRef
  4. P. Benvenuti and R. Mesiar: Pseudo-arithmetical operations as a basis for the general measure and integration theory. Inform. Sci. 160 (2004), 1-11.   DOI:10.1016/j.ins.2003.07.005
  5. A. M. Bica: One-sided fuzzy numbers and applications to integral equations from epidemiology. Fuzzy Sets and Systems 219 (2013), 27-48.   DOI:10.1016/j.fss.2012.08.002
  6. A. M. Bica: Algebraic structures for fuzzy numbers from categorial point of view. Soft Computing 11 (2007), 1099-1105.   DOI:10.1007/s00500-007-0167-x
  7. A. Bica: Categories and algebrical structures for real fuzzy numbers. Pure Math. Appl. 13 (2003), 1-2, 63-77.   CrossRef
  8. J. J. Buckley: On the algebra of interactive fuzzy numbers. Fuzzy Sets and Systems 32 (1989), 291-306.   DOI:10.1016/0165-0114(89)90261-3
  9. J. J. Buckley: On the algebra of interactive fuzzy numbers: the continuous case. Fuzzy Sets and Systems 37 (1990), 317-326.   DOI:10.1016/0165-0114(90)90029-6
  10. S. H. Chen: Operations of fuzzy numbers with step form membership function using function principle. Infor. Sci. 198 (1998), 149-155.   DOI:10.1016/s0020-0255(97)10070-6
  11. M. Daňková: Generalized extensionality of fuzzy relations. Fuzzy Sets and Systems 148 (2004), 291-304.   DOI:10.1016/j.fss.2004.02.011
  12. M. Daňková: Approximation of extensional fuzzy relations over a residuated lattice. Fuzzy Sets and Systems 161 (2010), 1973-1991.   DOI:10.1016/j.fss.2010.03.011
  13. M. Delgado, M. A. Vila and W. Voxman: On a canonical representation of fuzzy numbers. Fuzzy Sets and Systems 1998, 125-135.   DOI:10.1016/s0165-0114(96)00144-3
  14. D. Dubois and H. Prade: Fuzzy numbers: an overview. In: Analysis of fuzzy information (J. Bezdek, ed.), vol. f. CRS, Boca Raton 1987, pp. 3-39.   CrossRef
  15. D. Dubois and H. Prade: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York 1980.   CrossRef
  16. D. Dubois and H. Prade: Fuzzy real algebra: some results. Fuzzy Sets and Systems 2 (1979), 327-348.   DOI:10.1016/0165-0114(79)90005-8
  17. D. Dubois and H. Prade: Operations on fuzzy numbers. Int. J. Syst. Sci. 9 (1978), 613-626.   DOI:10.1080/00207727808941724
  18. R. Goetschel and W. Voxman: Elementary fuzzy calculus. Fuzzy Sets and Systems 18 (1986), 31-43.   DOI:10.1016/0165-0114(86)90026-6
  19. R. Goetschel and W. Voxman: Topological properties of fuzzy numbers. Fuzzy Sets and Systems 10 (1983), 87-99.   DOI:10.1016/s0165-0114(83)80107-9
  20. M. L. Guerra and L. Stefanini: Approximate fuzzy arithmetic operations using monotonic interpolations. Fuzzy Sets and Systems 150 (2005), 5-33.   DOI:10.1016/j.fss.2004.06.007
  21. M. Hanss: Applied Fuzzy Arithmetic: An Introduction with Engineering Applications. Springer-Verlag, Berlin 2005.   CrossRef
  22. M. Holčapek and M. Štěpnička: Arithmetics of extensional fuzzy numbers - Part I: Introduction. In: Proc. FUZZ-IEEE 2012, pp. 1517-1524.   DOI:10.1109/fuzz-ieee.2012.6251274
  23. M. Holčapek and M. Štěpnička: Arithmetics of extensional fuzzy numbers - Part II: Algebraic framework. In: Proc. FUZZ-IEEE 2012, pp. 1525-1532.   DOI:10.1109/fuzz-ieee.2012.6251275
  24. M. Holčapek, M. Wrublová and M. Štěpnička: On isomorphism theorems for MI-groups. In: Proc. EUSFLAT 2013, pp. 764-771.   CrossRef
  25. D. H. Hong and S. Lee: Some algebraic properties and a distance measure for interval-valued fuzzy numbers. Inform. Sci. 148 (2002), 1-10.   DOI:10.1016/s0020-0255(02)00265-7
  26. D. H. Hong, E. L. Moon and J. D. Kim: A note on the core of fuzzy numbers. Appl. Math. Lett. 23 (2010), 282-285.   DOI:10.1016/j.aml.2009.09.026
  27. M. Josephy: Composing functions of bounded variation. Proc. Amer. Math. Soc. 83 (1981), 354-356.   DOI:10.2307/2043527
  28. A. Kaufmann and M. M. Gupta: Introduction to Fuzzy Arithmetic. Van Nostrand Reinhold, New York 1991.   CrossRef
  29. W. Kosiński: Calculation and reasoning with ordered fuzzy numbers. In: EUSFLAT/LFA Proc. (E. Monseny and P. Sobrevilla, eds.), 2005, pp. 412-417.   CrossRef
  30. W. Kosiński, P. Prokopowicz and D. Ślezak: Calculus with fuzzy numbers. In: IMTCI 2004, LNAI 3490 (L. Bolc et all., eds.), Springer-Verlag, Berlin, Heidelberg 2005, pp. 21-28.   CrossRef
  31. W. Kosiński, P. Prokopowicz and D. Ślezak: Ordered fuzzy numbers. Bull. Pol. Acad. Sci. Math. 51 (2003), 3, 327-339.   CrossRef
  32. W. Kosiński: On fuzzy number calculus. Int. J. Appl. Math. Comput. Sci. 16 (2006), 51-57.   CrossRef
  33. M. Kovács and L. H. Tran: Algebraic structure of centered M-fuzzy numbers. Fuzzy Sets and Systems 39 (1991), 91-99.   DOI:10.1016/0165-0114(91)90068-2
  34. R. Lowen: On $\left( \mathbb{R}\left( L\right) ,\oplus\right) $. Fuzzy Sets and Systems. 10 (1983), 203-209.   DOI:10.1016/s0165-0114(83)80115-8
  35. M. Ma: On embedding problems of fuzzy number spaces: Part 4. Fuzzy Sets and Systems 58 (1993), 185-193.   DOI:10.1016/0165-0114(93)90494-3
  36. M. Ma, M. Friedman and A. Kandel: A new fuzzy arithmetic. Fuzzy Sets and Systems 108 (1999), 83-90.   DOI:10.1016/s0165-0114(97)00310-2
  37. Z. Makó: Real vector space of LR-fuzzy intervals with respect to the shape-preserving t-norm-based addition. Fuzzy Sets and Systems 200 (2012), 136-149.   DOI:10.1016/j.fss.2012.02.014
  38. M. Mareš: Addition of fuzzy quantities: disjunction-conjunction approach. Kybernetika 25 (1989), 2, 104-116.   CrossRef
  39. M. Mareš: Algebra of fuzzy quantities. Int. J. General Systems 20 (1991), 59-65.   DOI:10.1080/03081079108945014
  40. M. Mareš: Multiplication of fuzzy quantities. Kybernetika 28 (1992), 5, 337-356.   CrossRef
  41. M. Mareš: Algebraic equivalences over fuzzy quantities. Kybernetika 29 (1993), 2, 122-132.   CrossRef
  42. M. Mareš: Computation over Fuzzy Quantities. CRC Press, Boca Raton 1994.   CrossRef
  43. M. Mareš: Brief note on distributivity of triangular fuzzy numbers. Kybernetika 31 (1995), 5, 451-457.   CrossRef
  44. M. Mareš: Equivalentions over fuzzy quantities. Tatra Mountains Math. Publ. 6 (1995), 117-121.   CrossRef
  45. M. Mareš: Fuzzy zero, algebraic equivalence: yes or no? Kybernetika 32 (1996), 4, 343-351.   CrossRef
  46. M. Mareš: Weak arthmetics of fuzzy numbers. Fuzzy Sets and Systems 91 (1997), 143-153.   DOI:10.1016/s0165-0114(97)00136-x
  47. R. Mesiar and J. Rybárik: Pan-operations structure. Fuzzy Sets and Systems 74 (1995), 365-369.   DOI:10.1016/0165-0114(94)00314-w
  48. A. Maturo: On some structures of fuzzy numbers. Iranian J. Fuzzy Syst. 6 (2009), 49-59.   CrossRef
  49. I. P. Natanson: Theory of Functions of a Real Variable, 1-2. F. Ungar Publ. 1955 - 1961 (translated from Russian: Soviet Academic Press, Moscow 1950).   CrossRef
  50. D. Qiu, C. Lu, W. Zhang and Y. Lan: Algebraic properties and topological properties of the quotient space of fuzzy numbers based on Mareš equivalence relation. Fuzzy Sets and Systems 245 (2014), 63-82.   CrossRef
  51. D. Qiu and W. Zhang: Symmetric fuzzy numbers and additive equivalence of fuzzy numbers. Soft Computing 17 (2013), 1471-1477.   DOI:10.1007/s00500-013-1000-3
  52. H. L. Royden: Real Analysis. The Macmillan Co., New York, London 1969.   CrossRef
  53. L. Stefanini, L. Sorini and M. L. Guerra: Parametric representation of fuzzy numbers and application to fuzzy calculus. Fuzzy Sets and Systems 157 (2006), 2423-2455.   DOI:10.1016/j.fss.2006.02.002
  54. L. Stefanini: A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets and Systems 151 (2010), 1564-1584.   DOI:10.1016/j.fss.2009.06.009
  55. J. Vrba: A note on inverses in arithmetic with fuzzy numbers. Fuzzy Sets and Systems 50 (1992), 267-278.   DOI:10.1016/0165-0114(92)90225-s
  56. R. Zhao and R. Govind: Algebraic characteristics of extended fuzzy numbers. Inform. Sci. 54 (1991), 103-130.   DOI:10.1016/0020-0255(91)90047-x