Quotient algebraic structures on the set of fuzzy numbers. (English) Zbl 1349.22001
Summary: A. M. Bica has constructed in [Soft Comput. 11, No. 11, 1099–1105 (2007; Zbl 1125.03039)] 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 the 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:
function with bounded variation; semigroup (monoid) with involution; topological group; metric spaceCitations:
Zbl 1125.03039Software:
FUSIMReferences:
| [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. · Zbl 1104.26022 |
| [2] | Ban, A. I., Bica, A.: Solving systems of equivalentions. J. Appl. Math. Comput. 20 (2006), 1-2, 97-118. · Zbl 1104.15005 · doi:10.1007/BF02831926 |
| [3] | Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic. Springer-Verlag, Berlin 2013. · Zbl 1271.03001 · doi:10.1007/978-3-642-35221-8 |
| [4] | Benvenuti, P., Mesiar, R.: Pseudo-arithmetical operations as a basis for the general measure and integration theory. Inform. Sci. 160 (2004), 1-11. · Zbl 1040.28020 · doi:10.1016/j.ins.2003.07.005 |
| [5] | Bica, A. M.: One-sided fuzzy numbers and applications to integral equations from epidemiology. Fuzzy Sets and Systems 219 (2013), 27-48. · Zbl 1276.92098 · doi:10.1016/j.fss.2012.08.002 |
| [6] | Bica, A. M.: Algebraic structures for fuzzy numbers from categorial point of view. Soft Computing 11 (2007), 1099-1105. · Zbl 1125.03039 · doi:10.1007/s00500-007-0167-x |
| [7] | Bica, A.: Categories and algebrical structures for real fuzzy numbers. Pure Math. Appl. 13 (2003), 1-2, 63-77. · Zbl 1026.26024 |
| [8] | Buckley, J. J.: On the algebra of interactive fuzzy numbers. Fuzzy Sets and Systems 32 (1989), 291-306. · Zbl 0708.04005 · doi:10.1016/0165-0114(90)90029-6 |
| [9] | Buckley, J. J.: On the algebra of interactive fuzzy numbers: the continuous case. Fuzzy Sets and Systems 37 (1990), 317-326. · Zbl 0708.04005 · doi:10.1016/0165-0114(90)90029-6 |
| [10] | Chen, S. H.: Operations of fuzzy numbers with step form membership function using function principle. Infor. Sci. 198 (1998), 149-155. · Zbl 0922.04007 · doi:10.1016/S0020-0255(97)10070-6 |
| [11] | Daňková, M.: Generalized extensionality of fuzzy relations. Fuzzy Sets and Systems 148 (2004), 291-304. · Zbl 1059.03057 · doi:10.1016/j.fss.2004.02.011 |
| [12] | Daňková, M.: Approximation of extensional fuzzy relations over a residuated lattice. Fuzzy Sets and Systems 161 (2010), 1973-1991. · Zbl 1205.03036 · doi:10.1016/j.fss.2010.03.011 |
| [13] | Delgado, M., Vila, M. A., Voxman, W.: On a canonical representation of fuzzy numbers. Fuzzy Sets and Systems 1998, 125-135. · Zbl 0916.04004 · doi:10.1016/S0165-0114(96)00144-3 |
| [14] | Dubois, D., Prade, H.: Fuzzy numbers: an overview. Analysis of fuzzy information (J. Bezdek, vol. f. CRS, Boca Raton 1987, pp. 3-39. · Zbl 0663.94028 |
| [15] | Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York 1980. · Zbl 0444.94049 |
| [16] | Dubois, D., Prade, H.: Fuzzy real algebra: some results. Fuzzy Sets and Systems 2 (1979), 327-348. · Zbl 0412.03035 · doi:10.1016/0165-0114(79)90005-8 |
| [17] | Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9 (1978), 613-626. · Zbl 0383.94045 · doi:10.1080/00207727808941724 |
| [18] | Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets and Systems 18 (1986), 31-43. · Zbl 0626.26014 · doi:10.1016/0165-0114(86)90026-6 |
| [19] | Goetschel, R., Voxman, W.: Topological properties of fuzzy numbers. Fuzzy Sets and Systems 10 (1983), 87-99. · Zbl 0521.54001 · doi:10.1016/S0165-0114(83)80107-9 |
| [20] | Guerra, M. L., Stefanini, L.: Approximate fuzzy arithmetic operations using monotonic interpolations. Fuzzy Sets and Systems 150 (2005), 5-33. · Zbl 1062.03050 · doi:10.1016/j.fss.2004.06.007 |
| [21] | Hanss, M.: Applied Fuzzy Arithmetic: An Introduction with Engineering Applications. Springer-Verlag, Berlin 2005. · Zbl 1085.03041 |
| [22] | Holčapek, M., Štěpnička, M.: Arithmetics of extensional fuzzy numbers - Part I: Introduction. Proc. FUZZ-IEEE 2012, pp. 1517-1524. · doi:10.1109/fuzz-ieee.2012.6251274 |
| [23] | Holčapek, M., Štěpnička, M.: Arithmetics of extensional fuzzy numbers - Part II: Algebraic framework. Proc. FUZZ-IEEE 2012, pp. 1525-1532. · doi:10.1109/fuzz-ieee.2012.6251275 |
| [24] | Holčapek, M., Wrublová, M., Štěpnička, M.: On isomorphism theorems for MI-groups. Proc. EUSFLAT 2013, pp. 764-771. |
| [25] | Hong, D. H., Lee, S.: Some algebraic properties and a distance measure for interval-valued fuzzy numbers. Inform. Sci. 148 (2002), 1-10. · Zbl 1019.03036 · doi:10.1016/S0020-0255(02)00265-7 |
| [26] | Hong, D. H., Moon, E. L., Kim, J. D.: A note on the core of fuzzy numbers. Appl. Math. Lett. 23 (2010), 282-285. · Zbl 1185.26062 · doi:10.1016/j.aml.2009.09.026 |
| [27] | Josephy, M.: Composing functions of bounded variation. Proc. Amer. Math. Soc. 83 (1981), 354-356. · Zbl 0475.26005 · doi:10.2307/2043527 |
| [28] | Kaufmann, A., Gupta, M. M.: Introduction to Fuzzy Arithmetic. Van Nostrand Reinhold, New York 1991. · Zbl 0754.26012 |
| [29] | Kosiński, W.: Calculation and reasoning with ordered fuzzy numbers. EUSFLAT/LFA Proc. (E. Monseny and P. Sobrevilla, 2005, pp. 412-417. |
| [30] | Kosiński, W., Prokopowicz, P., Ślezak, D.: Calculus with fuzzy numbers. IMTCI 2004, LNAI 3490 (L. Bolc et all., 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. · Zbl 1102.03310 |
| [32] | Kosiński, W.: On fuzzy number calculus. Int. J. Appl. Math. Comput. Sci. 16 (2006), 51-57. · Zbl 1334.94112 |
| [33] | Kovács, M., Tran, L. H.: Algebraic structure of centered M-fuzzy numbers. Fuzzy Sets and Systems 39 (1991), 91-99. · Zbl 0724.04006 · doi:10.1016/0165-0114(91)90068-2 |
| [34] | Lowen, R.: On \(\left( \mathbb{R}\left( L\right) ,\oplus\right) \). Fuzzy Sets and Systems. 10 (1983), 203-209. · Zbl 0527.54006 · doi:10.1016/s0165-0114(83)80115-8 |
| [35] | Ma, M.: On embedding problems of fuzzy number spaces: Part 4. Fuzzy Sets and Systems 58 (1993), 185-193. · Zbl 0789.46069 · doi:10.1016/0165-0114(93)90494-3 |
| [36] | Ma, M., Friedman, M., Kandel, A.: A new fuzzy arithmetic. Fuzzy Sets and Systems 108 (1999), 83-90. · Zbl 0937.03059 · doi:10.1016/S0165-0114(97)00310-2 |
| [37] | Makó, Z.: 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. · Zbl 1251.15002 · doi:10.1016/j.fss.2012.02.014 |
| [38] | Mareš, M.: Addition of fuzzy quantities: disjunction-conjunction approach. Kybernetika 25 (1989), 2, 104-116. · Zbl 0679.90027 |
| [39] | Mareš, M.: Algebra of fuzzy quantities. Int. J. General Systems 20 (1991), 59-65. · Zbl 0739.90074 · doi:10.1080/03081079108945014 |
| [40] | Mareš, M.: Multiplication of fuzzy quantities. Kybernetika 28 (1992), 5, 337-356. · Zbl 0786.04006 |
| [41] | Mareš, M.: Algebraic equivalences over fuzzy quantities. Kybernetika 29 (1993), 2, 122-132. · Zbl 0788.04006 |
| [42] | Mareš, M.: Computation over Fuzzy Quantities. CRC Press, Boca Raton 1994. · Zbl 0859.94035 |
| [43] | Mareš, M.: Brief note on distributivity of triangular fuzzy numbers. Kybernetika 31 (1995), 5, 451-457. · Zbl 0856.04009 |
| [44] | Mareš, M.: Equivalentions over fuzzy quantities. Tatra Mountains Math. Publ. 6 (1995), 117-121. · Zbl 0849.04005 |
| [45] | Mareš, M.: Fuzzy zero, algebraic equivalence: yes or no?. Kybernetika 32 (1996), 4, 343-351. · Zbl 0884.04004 |
| [46] | Mareš, M.: Weak arthmetics of fuzzy numbers. Fuzzy Sets and Systems 91 (1997), 143-153. · Zbl 0919.04009 · doi:10.1016/s0165-0114(97)00136-x |
| [47] | Mesiar, R., Rybárik, J.: Pan-operations structure. Fuzzy Sets and Systems 74 (1995), 365-369. · Zbl 0859.28010 · doi:10.1016/0165-0114(94)00314-W |
| [48] | Maturo, A.: On some structures of fuzzy numbers. Iranian J. Fuzzy Syst. 6 (2009), 49-59. · Zbl 1203.03084 |
| [49] | Natanson, I. P.: Theory of Functions of a Real Variable, 1-2. F. Ungar Publ. 1955 - 1961 ( · Zbl 0064.29102 |
| [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. · Zbl 1315.03100 · doi:10.1016/j.fss.2014.01.003 |
| [51] | Qiu, D., Zhang, W.: Symmetric fuzzy numbers and additive equivalence of fuzzy numbers. Soft Computing 17 (2013), 1471-1477. · Zbl 1325.03064 · doi:10.1007/s00500-013-1000-3 |
| [52] | Royden, H. L.: Real Analysis. The Macmillan Co., New York, London 1969. · Zbl 1191.26002 |
| [53] | Stefanini, L., Sorini, L., Guerra, M. L.: Parametric representation of fuzzy numbers and application to fuzzy calculus. Fuzzy Sets and Systems 157 (2006), 2423-2455. · Zbl 1109.26024 · doi:10.1016/j.fss.2006.02.002 |
| [54] | Stefanini, L.: A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets and Systems 151 (2010), 1564-1584. · Zbl 1188.26019 · doi:10.1016/j.fss.2009.06.009 |
| [55] | Vrba, J.: 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] | Zhao, R., Govind, R.: Algebraic characteristics of extended fuzzy numbers. Inform. Sci. 54 (1991), 103-130. · Zbl 0774.26015 · doi:10.1016/0020-0255(91)90047-X |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
