Quotient algebraic structures on the set of fuzzy numbers

Dorina Fechete; Ioan Fechete

Kybernetika (2015)

  • Volume: 51, Issue: 2, page 255-267
  • ISSN: 0023-5954

Abstract

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A. M. Bica has constructed in [6] 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 ] .

How to cite

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Fechete, Dorina, and Fechete, Ioan. "Quotient algebraic structures on the set of fuzzy numbers." Kybernetika 51.2 (2015): 255-267. <http://eudml.org/doc/270136>.

@article{Fechete2015,
abstract = {A. M. Bica has constructed in [6] 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 $[0,1]$.},
author = {Fechete, Dorina, Fechete, Ioan},
journal = {Kybernetika},
keywords = {fuzzy number; function with bounded variation; semigroup (monoid) with involution; topological group; metric space; fuzzy number; function with bounded variation; semigroup (monoid) with involution; topological group; metric space},
language = {eng},
number = {2},
pages = {255-267},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Quotient algebraic structures on the set of fuzzy numbers},
url = {http://eudml.org/doc/270136},
volume = {51},
year = {2015},
}

TY - JOUR
AU - Fechete, Dorina
AU - Fechete, Ioan
TI - Quotient algebraic structures on the set of fuzzy numbers
JO - Kybernetika
PY - 2015
PB - Institute of Information Theory and Automation AS CR
VL - 51
IS - 2
SP - 255
EP - 267
AB - A. M. Bica has constructed in [6] 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 $[0,1]$.
LA - eng
KW - fuzzy number; function with bounded variation; semigroup (monoid) with involution; topological group; metric space; fuzzy number; function with bounded variation; semigroup (monoid) with involution; topological group; metric space
UR - http://eudml.org/doc/270136
ER -

References

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  1. Ban, A. I., Bede, B., Power series of fuzzy numbers withy cross product and applications to fuzzy differential equations., J. Concrete Appl. Math 4 (2006), 2, 125-152. MR2220289
  2. Ban, A. I., Bica, A., 10.1007/bf02831926, J. Appl. Math. Comput. 20 (2006), 1-2, 97-118. Zbl1104.15005MR2188352DOI10.1007/bf02831926
  3. Bede, B., Mathematics of Fuzzy Sets and Fuzzy Logic., Springer-Verlag, Berlin 2013. Zbl1271.03001MR3024762
  4. Benvenuti, P., Mesiar, R., 10.1016/j.ins.2003.07.005, Inform. Sci. 160 (2004), 1-11. Zbl1040.28020MR2041855DOI10.1016/j.ins.2003.07.005
  5. Bica, A. M., 10.1016/j.fss.2012.08.002, Fuzzy Sets and Systems 219 (2013), 27-48. Zbl1276.92098MR3035732DOI10.1016/j.fss.2012.08.002
  6. Bica, A. M., 10.1007/s00500-007-0167-x, Soft Computing 11 (2007), 1099-1105. Zbl1125.03039DOI10.1007/s00500-007-0167-x
  7. Bica, A., Categories and algebrical structures for real fuzzy numbers., Pure Math. Appl. 13 (2003), 1-2, 63-77. MR1987199
  8. Buckley, J. J., 10.1016/0165-0114(89)90261-3, Fuzzy Sets and Systems 32 (1989), 291-306. Zbl0708.04005MR1018719DOI10.1016/0165-0114(89)90261-3
  9. Buckley, J. J., 10.1016/0165-0114(90)90029-6, Fuzzy Sets and Systems 37 (1990), 317-326. Zbl0708.04005MR1077148DOI10.1016/0165-0114(90)90029-6
  10. Chen, S. H., 10.1016/s0020-0255(97)10070-6, Infor. Sci. 198 (1998), 149-155. Zbl0922.04007MR1632503DOI10.1016/s0020-0255(97)10070-6
  11. Daňková, M., 10.1016/j.fss.2004.02.011, Fuzzy Sets and Systems 148 (2004), 291-304. Zbl1059.03057MR2100201DOI10.1016/j.fss.2004.02.011
  12. Daňková, M., 10.1016/j.fss.2010.03.011, Fuzzy Sets and Systems 161 (2010), 1973-1991. Zbl1205.03036MR2639424DOI10.1016/j.fss.2010.03.011
  13. Delgado, M., Vila, M. A., Voxman, W., 10.1016/s0165-0114(96)00144-3, Fuzzy Sets and Systems 1998, 125-135. Zbl0916.04004DOI10.1016/s0165-0114(96)00144-3
  14. Dubois, D., Prade, H., Fuzzy numbers: an overview., In: Analysis of fuzzy information (J. Bezdek, ed.), vol. f. CRS, Boca Raton 1987, pp. 3-39. Zbl0663.94028MR0910312
  15. Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications., Academic Press, New York 1980. Zbl0444.94049MR0700429
  16. Dubois, D., Prade, H., 10.1016/0165-0114(79)90005-8, Fuzzy Sets and Systems 2 (1979), 327-348. Zbl0412.03035MR0545836DOI10.1016/0165-0114(79)90005-8
  17. Dubois, D., Prade, H., 10.1080/00207727808941724, Int. J. Syst. Sci. 9 (1978), 613-626. Zbl0383.94045MR0491199DOI10.1080/00207727808941724
  18. Goetschel, R., Voxman, W., 10.1016/0165-0114(86)90026-6, Fuzzy Sets and Systems 18 (1986), 31-43. Zbl0626.26014MR0825618DOI10.1016/0165-0114(86)90026-6
  19. Goetschel, R., Voxman, W., 10.1016/s0165-0114(83)80107-9, Fuzzy Sets and Systems 10 (1983), 87-99. Zbl0521.54001MR0700437DOI10.1016/s0165-0114(83)80107-9
  20. Guerra, M. L., Stefanini, L., 10.1016/j.fss.2004.06.007, Fuzzy Sets and Systems 150 (2005), 5-33. Zbl1062.03050MR2113512DOI10.1016/j.fss.2004.06.007
  21. Hanss, M., Applied Fuzzy Arithmetic: An Introduction with Engineering Applications., Springer-Verlag, Berlin 2005. Zbl1085.03041
  22. Holčapek, M., Štěpnička, M., 10.1109/fuzz-ieee.2012.6251274, In: Proc. FUZZ-IEEE 2012, pp. 1517-1524. DOI10.1109/fuzz-ieee.2012.6251274
  23. Holčapek, M., Štěpnička, M., 10.1109/fuzz-ieee.2012.6251275, In: Proc. FUZZ-IEEE 2012, pp. 1525-1532. DOI10.1109/fuzz-ieee.2012.6251275
  24. Holčapek, M., Wrublová, M., Štěpnička, M., On isomorphism theorems for MI-groups., In: Proc. EUSFLAT 2013, pp. 764-771. 
  25. Hong, D. H., Lee, S., 10.1016/s0020-0255(02)00265-7, Inform. Sci. 148 (2002), 1-10. Zbl1019.03036MR1947110DOI10.1016/s0020-0255(02)00265-7
  26. Hong, D. H., Moon, E. L., Kim, J. D., 10.1016/j.aml.2009.09.026, Appl. Math. Lett. 23 (2010), 282-285. Zbl1185.26062MR2565191DOI10.1016/j.aml.2009.09.026
  27. Josephy, M., 10.2307/2043527, Proc. Amer. Math. Soc. 83 (1981), 354-356. Zbl0475.26005MR0624930DOI10.2307/2043527
  28. Kaufmann, A., Gupta, M. M., Introduction to Fuzzy Arithmetic., Van Nostrand Reinhold, New York 1991. Zbl0754.26012MR1132439
  29. Kosiński, W., Calculation and reasoning with ordered fuzzy numbers., In: EUSFLAT/LFA Proc. (E. Monseny and P. Sobrevilla, eds.), 2005, pp. 412-417. 
  30. Kosiński, W., Prokopowicz, P., Ślezak, D., Calculus with fuzzy numbers., In: IMTCI 2004, LNAI 3490 (L. Bolc et all., eds.), Springer-Verlag, Berlin, Heidelberg 2005, pp. 21-28. 
  31. Kosiński, W., Prokopowicz, P., Ślezak, D., Ordered fuzzy numbers., Bull. Pol. Acad. Sci. Math. 51 (2003), 3, 327-339. Zbl1102.03310MR2017009
  32. Kosiński, W., On fuzzy number calculus., Int. J. Appl. Math. Comput. Sci. 16 (2006), 51-57. MR2212158
  33. Kovács, M., Tran, L. H., 10.1016/0165-0114(91)90068-2, Fuzzy Sets and Systems 39 (1991), 91-99. Zbl0724.04006MR1089014DOI10.1016/0165-0114(91)90068-2
  34. Lowen, R., 10.1016/s0165-0114(83)80115-8, Fuzzy Sets and Systems. 10 (1983), 203-209. MR0705208DOI10.1016/s0165-0114(83)80115-8
  35. Ma, M., 10.1016/0165-0114(93)90494-3, Fuzzy Sets and Systems 58 (1993), 185-193. MR1239476DOI10.1016/0165-0114(93)90494-3
  36. Ma, M., Friedman, M., Kandel, A., 10.1016/s0165-0114(97)00310-2, Fuzzy Sets and Systems 108 (1999), 83-90. Zbl0937.03059MR1714662DOI10.1016/s0165-0114(97)00310-2
  37. Makó, Z., 10.1016/j.fss.2012.02.014, Fuzzy Sets and Systems 200 (2012), 136-149. Zbl1251.15002MR2927849DOI10.1016/j.fss.2012.02.014
  38. Mareš, M., Addition of fuzzy quantities: disjunction-conjunction approach., Kybernetika 25 (1989), 2, 104-116. Zbl0679.90027MR0995953
  39. Mareš, M., 10.1080/03081079108945014, Int. J. General Systems 20 (1991), 59-65. Zbl0739.90074DOI10.1080/03081079108945014
  40. Mareš, M., Multiplication of fuzzy quantities., Kybernetika 28 (1992), 5, 337-356. Zbl0786.04006MR1197719
  41. Mareš, M., Algebraic equivalences over fuzzy quantities., Kybernetika 29 (1993), 2, 122-132. Zbl0788.04006MR1227746
  42. Mareš, M., Computation over Fuzzy Quantities., CRC Press, Boca Raton 1994. Zbl0859.94035MR1327525
  43. Mareš, M., Brief note on distributivity of triangular fuzzy numbers., Kybernetika 31 (1995), 5, 451-457. Zbl0856.04009MR1361306
  44. Mareš, M., Equivalentions over fuzzy quantities., Tatra Mountains Math. Publ. 6 (1995), 117-121. Zbl0849.04005MR1363989
  45. Mareš, M., Fuzzy zero, algebraic equivalence: yes or no?, Kybernetika 32 (1996), 4, 343-351. Zbl0884.04004MR1420127
  46. Mareš, M., 10.1016/s0165-0114(97)00136-x, Fuzzy Sets and Systems 91 (1997), 143-153. MR1480041DOI10.1016/s0165-0114(97)00136-x
  47. Mesiar, R., Rybárik, J., 10.1016/0165-0114(94)00314-w, Fuzzy Sets and Systems 74 (1995), 365-369. Zbl0859.28010MR1351585DOI10.1016/0165-0114(94)00314-w
  48. Maturo, A., On some structures of fuzzy numbers., Iranian J. Fuzzy Syst. 6 (2009), 49-59. Zbl1203.03084MR2604175
  49. Natanson, I. P., Theory of Functions of a Real Variable, 1-2., F. Ungar Publ. 1955 - 1961 (translated from Russian: Soviet Academic Press, Moscow 1950). MR0039790
  50. Qiu, D., Lu, C., Zhang, W., Lan, Y., 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. MR3195426
  51. Qiu, D., Zhang, W., 10.1007/s00500-013-1000-3, Soft Computing 17 (2013), 1471-1477. DOI10.1007/s00500-013-1000-3
  52. Royden, H. L., Real Analysis., The Macmillan Co., New York, London 1969. Zbl1191.26002MR1013117
  53. Stefanini, L., Sorini, L., Guerra, M. L., 10.1016/j.fss.2006.02.002, Fuzzy Sets and Systems 157 (2006), 2423-2455. Zbl1109.26024MR2254174DOI10.1016/j.fss.2006.02.002
  54. Stefanini, L., 10.1016/j.fss.2009.06.009, Fuzzy Sets and Systems 151 (2010), 1564-1584. Zbl1188.26019MR2608262DOI10.1016/j.fss.2009.06.009
  55. Vrba, J., 10.1016/0165-0114(92)90225-s, Fuzzy Sets and Systems 50 (1992), 267-278. MR1188297DOI10.1016/0165-0114(92)90225-s
  56. Zhao, R., Govind, R., 10.1016/0020-0255(91)90047-x, Inform. Sci. 54 (1991), 103-130. Zbl0774.26015MR1079201DOI10.1016/0020-0255(91)90047-x

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