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Symmetric fuzzy numbers and additive equivalence of fuzzy numbers

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Abstract

To obtain the group properties of fuzzy quantities, Mareš introduced an equivalence relation between fuzzy quantities. However, the Mareš’s method used to prove his main theorems demands to limit the investigation to fuzzy quantities with finite support. In this paper, we discuss the properties of symmetric fuzzy numbers, show an equivalent characterization of convex fuzzy sets, and present a way to construct a symmetric convex fuzzy set with a convex fuzzy set. Based on these results, we restrict ourselves to fuzzy numbers and prove Mareš’s results without the limitation to fuzzy numbers with finite support using the refined equivalence relation due to Hong and Do. Our results prove one of Mareš’s open questions in the literature.

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References

  • Anastassiou GA (2002) On H-fuzzy differentiation. Math Balkanica 16:155–193

    MathSciNet  Google Scholar 

  • Aytar S (2004) Statistical limit points of sequences of fuzzy numbers. Inf Sci 165:129–138

    Article  MathSciNet  MATH  Google Scholar 

  • Báez-Sánchez AD, Moretti AC, Rojas-Medar MA (2012) On polygonal fuzzy sets and numbers. Fuzzy Sets Syst. doi:10.1016/j.fss.2012.04.003

  • Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151:581–599

    Article  MathSciNet  MATH  Google Scholar 

  • Bica AM (2007) Algebraic structures for fuzzy number from categorial point of view. Soft Comput 11:1099–1105

    Article  MATH  Google Scholar 

  • Diamond P, Kloeden P (1990) Metric spaces of fuzzy sets. Fuzzy Sets Syst 35:241–249

    Article  MathSciNet  MATH  Google Scholar 

  • Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets: theory and applications. World Scientific, Singapore

  • Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D, Prade H (1980) Fuzzy sets and systems. Academic Press, New York

  • Dubois D, Prade H (1981) Additions of interactive fuzzy numbers. IEEE Trans Autom Control 26:926–936

    Article  MathSciNet  Google Scholar 

  • Dubois D, Prade H (1984) Fuzzy-set-theoretic differences and inclusions and their use in the analysis of fuzzy equations. Control Cybernet (Warsaw) 13:129–146

    MathSciNet  MATH  Google Scholar 

  • Dubois D, Prade H (1987) Fuzzy numbers: an overview. In: Bezdek J (ed) Analysis of fuzzy information, vol f. CRS, Boca Raton, pp 3–39

  • Höhle U (1988) Quotients with respect to similarity relations. Fuzzy Sets Syst 27:31–44

    Article  MATH  Google Scholar 

  • Hong D, Do H (1996) Additive decomposition of fuzzy quantities. Inf Sci 88:201–207

    Article  MathSciNet  MATH  Google Scholar 

  • Hong D (2003) An additive decomposition of fuzzy numbers. Kybernetika 39:289–294

    MathSciNet  MATH  Google Scholar 

  • Hong D, Lee S (2002) Some algebraic properties and a distance measure for interval-valued fuzzy numbers. Inf Sci 148:1–10

    Article  MathSciNet  MATH  Google Scholar 

  • Hukuhara M (1967) Integration des applications measurables dont la valeur est un compact convexe. Funkcialaj Ekvacioj 10:205–223

    MathSciNet  MATH  Google Scholar 

  • Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317

    Article  MathSciNet  MATH  Google Scholar 

  • Kovács M (1992) A stable embedding of ill-posed linear systems into fuzzy systems. Fuzzy Sets Syst 45:305–312

    Article  MATH  Google Scholar 

  • Makó Z (2012) Real vector space of LR-fuzzy intervals with respect to the shape-preserving t-norm-based addition. Fuzzy Sets Syst 200:136–149

    Article  MATH  Google Scholar 

  • Mareš M (1983) Fuzzy quantities in networks. Fuzzy Sets Syst 10:123–134

    Article  MATH  Google Scholar 

  • Mareš M (1988) How to satisfy fuzzy manager. Ekonomickomatematicky Obzor 24:396–404

    MATH  Google Scholar 

  • Mareš M (1989) Addition of fuzzy quantities: disjunction-conjunction approach. Kybernetika 25:104–116

    MathSciNet  MATH  Google Scholar 

  • Mareš M (1992) Additive decomposition of fuzzy quantities. Fuzzy Sets Syst 47:341–346

    Article  MATH  Google Scholar 

  • Maturo A (2009) On some structures of fuzzy numbers. Iranian J Fuzzy Syst 6:49–59

    MathSciNet  MATH  Google Scholar 

  • Mesiar R (1996) Anote to the T-sum of L-R fuzzy numbers. Fuzzy Sets Syst 79:259–261

    Article  MathSciNet  MATH  Google Scholar 

  • Michta M (2011) On set-valued stochastic integrals and fuzzy stochastic equations. Fuzzy Sets Syst 177:1–19

    Article  MathSciNet  MATH  Google Scholar 

  • Negoita CV, Ralescu D (1975) Representation theorems for fuzzy concepts. Kybernetes 4:169–174

    Article  MATH  Google Scholar 

  • Panda G, Panigrahi M, Nanda S (2006) Equivalence class in the set of fuzzy numbers and its application in decision-making problems. Int J Math Math Sci. Article ID 74165

  • Pushkov SG (2008) Fuzzy modules with respect to t-norm and some of their properties. J Math Sci 154:374–378

    Article  MathSciNet  MATH  Google Scholar 

  • Qiu D, Shu L (2008) Notes on “On the restudy of fuzzy complex analysis: part I and part II". Fuzzy Sets Syst 159:2185–2189

    Article  MathSciNet  MATH  Google Scholar 

  • Qiu D, Shu L, Mo Z (2009a) Notes on fuzzy complex analysis. Fuzzy Sets Syst 160:1578–1589

    Article  MathSciNet  MATH  Google Scholar 

  • Qiu D, Shu L, Mo Z (2009b) On starshaped fuzzy sets. Fuzzy Sets Syst 160:1563–1577

    Article  MathSciNet  MATH  Google Scholar 

  • Qiu D, Yang F, Shu L (2010) On convex fuzzy processes and their generalizations. Int J Fuzzy Syst 12:268–273

    MathSciNet  Google Scholar 

  • Stefanini L (2010) A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161:1564–1584

    Article  MathSciNet  MATH  Google Scholar 

  • Stefanini L, Sorini L, Guerra ML (2006) Paremetric representation of fuzzy numbers and application to fuzzy calculus. Fuzzy Sets Syst 157:2423–2455

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Baruah A (2009) New type of difference sequence spaces of fuzzy real numbers. Math Model Anal 14:391–397

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Baruah A (2010a) Norlund and Riesz mean of sequences of fuzzy real numbers. Appl Math Lett 23:651–655

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Baruah A (2010b) Lacunary statistically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers. Kyungpook Math J 50:565–574

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Borgogain S (2008) The sequence space m(M, ϕ, ▵ m n , p)F. Math Model Anal 13:577–586

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Borgogain S (2011) Some classes of difference sequence spaces of fuzzy real numbers defined by Orlicz function. Adv Fuzzy Syst. Article ID 216414

  • Tripathy BC, Dutta AJ (2007) On fuzzy real-valued double sequence spaces. Math Comput Model 46:1294–1299

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Dutta AJ (2010) Bounded variation double sequence space of fuzzy real numbers. Comput Math Appl 59:1031–1037

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Sarma B (2008) Sequence spaces of fuzzy real numbers defined by Orlicz functions. Mathematica Slovaca 58:621–628

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Sarma B (2011) Double sequence spaces of fuzzy numbers defined by Orlicz function. Acta Mathematica Scientia 31:134–140

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy BC, Sarma B (2012) On I-convergent double sequences of fuzzy real numbers. Kyungpook Math J 52:189–200

    Article  MathSciNet  Google Scholar 

  • Tripathy BC, Sen M, Nath S (2012) I-convergence in probabilistic n-normed space. Soft Comput 16:1021–1027

    Article  MATH  Google Scholar 

  • Wang G, Li Y, Wen C (2007) On fuzzy n-cell numbers and n-dimension fuzzy vectors. Fuzzy Sets Syst 158:71–84

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh LA (1975) The concept of alinguistic variable and its applications to approximate reasoning. Parts I, II, III. Inf Sci 8:199–251, 301–357; 9:43–80

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Acknowledgments

The authors thank the anonymous reviewers for their valuable comments. This work was supported by The Mathematical Tianyuan Foundation of China (Grant no. 11126087), The National Natural Science Foundation of China (Grant no.11201512), and The Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001).

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Authors and Affiliations

  1. College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Nanan, Chongqing, 400065, People’s Republic of China

    Dong Qiu

  2. School of Information Engineering, Guangdong Medical College, Dongguan, Guangdong, 523808, People’s Republic of China

    Weiquan Zhang

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  1. Dong Qiu
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  2. Weiquan Zhang
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Correspondence to Dong Qiu.

Additional information

Communicated by A. Di Nola.

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Qiu, D., Zhang, W. Symmetric fuzzy numbers and additive equivalence of fuzzy numbers. Soft Comput 17, 1471–1477 (2013). https://doi.org/10.1007/s00500-013-1000-3

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