ORIGINAL_ARTICLE
Cover vol. 13, no. 1, February 2016
http://ijfs.usb.ac.ir/article_2631_dc98dadae6fef4885991297e38de7d91.pdf
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10.22111/ijfs.2016.2631
ORIGINAL_ARTICLE
Arithmetic Aggregation Operators for Interval-valued Intuitionistic Linguistic Variables and Application to Multi-attribute Group Decision Making
The intuitionistic linguistic set (ILS) is an extension of linguisitc variable. To overcome the drawback of using single real number to represent membership degree and non-membership degree for ILS, the concept of interval-valued intuitionistic linguistic set (IVILS) is introduced through representing the membership degree and non-membership degree with intervals for ILS in this paper. The operation law, score function, accuracy function , and certainty function for interval-valued intuitionistic linguistic varibales (IVILVs) are defined. Hereby a lexicographic method is proposed to rank the IVILVs. Then, three kinds of interval-valued intuitionistic linguistic arithmetic average operators are defined, including the interval-valued intuitionistic linguistic weighted arithmetic average (IVILWAA) operator, interval-valued intuitionistic linguistic ordered weighted arithmetic (IVILOWA) operator, and interval-valued intuitionistic linguistic hybrid arithmetic (IVILHA) operator, and their desirable properties are also discussed. Based on the IVILWAA and IVILHA operators, two methods are proposed for solving multi-attribute group decision making problems with IVILVs. Finally, an investment selection example is illustrated to demonstrate the applicability and validity of the methods proposed in this paper.
http://ijfs.usb.ac.ir/article_2284_a8c89f45da6d004df613f2dedf70b6f9.pdf
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23
10.22111/ijfs.2016.2284
Multi-attribute group decision making
Interval-valued intuitionistic linguistic set
Intuitionistic fuzzy set
Arithmetic operators
Jiuying
Dong
jiuyingdong@126.com
true
1
School of Statistics, Jiangxi University of Finance and Economics,
Nanchang 330013, China and Research Center of Applied Statistics, Jiangxi University
of Finance and Economics, Nanchang 330013, China
School of Statistics, Jiangxi University of Finance and Economics,
Nanchang 330013, China and Research Center of Applied Statistics, Jiangxi University
of Finance and Economics, Nanchang 330013, China
School of Statistics, Jiangxi University of Finance and Economics,
Nanchang 330013, China and Research Center of Applied Statistics, Jiangxi University
of Finance and Economics, Nanchang 330013, China
AUTHOR
Shu-Ping
Wan
true
2
College of Information Technology, Jiangxi University of Finance
and Economics, Nanchang 330013, China
College of Information Technology, Jiangxi University of Finance
and Economics, Nanchang 330013, China
College of Information Technology, Jiangxi University of Finance
and Economics, Nanchang 330013, China
LEAD_AUTHOR
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1
[2] K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and
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Systems, 31 (3) (1989), 343-349.
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[3] D. P. Filev and R. R. Yager, On the issue of obtaining OWA operator weights, Fuzzy Sets
4
and Systems, 94(2) (1998), 157-169.
5
[4] F. Herrera and L. Martnez, A 2-tuple fuzzy linguistic representation model for computing
6
with words, IEEE Transactions on Fuzzy Systems, 8 (2000), 746-752.
7
[5] F. Herrera, L. Martnez and P. J. Snchez, Managing non-homogeneous information in group
8
decision-making, European Journal of Operational Research, 166(1) (2005), 115-132.
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[6] P. D. Liu and F. Jin, Methods for aggregating intuitionistic uncertain linguistic variables
10
and their application to group decision making, Information Sciences, 205 (2012), 58-71.
11
[7] J. M. Merigo, M. Casanovas and L. Martnez, Linguistic aggregation operators for linguistic
12
decision making based on the Dempster-Shafer theory of evidence, International Journal
13
of Uncertainty, Fuzziness and Knowledge-Based Systems, 18(3) (2010), 287-304.
14
[8] J. M. Merigo, A unied model between the weighted average and the induced OWA oper-
15
ator, Expert Systems with Applications, 38(9) (2011), 11560-11572.
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[9] J. M. Merigo and A. M. Gil-Lafuente, Decision making techniques in business and econom-
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ics based on the OWA operator, SORT C Statistics and Operations Research Transactions,
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36(1) (2012), 81-102.
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[10] J. M. Merigo, A. M. Gil-Lafuente, L. G. Zhou and H. Y. Chen, Induced and linguistic
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generalized aggregation operators and their application in linguistic group decision making,
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Group Decision and Negotiation, 21 (2012), 531-549.
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set logic. In: Proc 22nd Annual IEEE Asilomar Conference on Signals, Systems and Computers.
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Pacic Grove, CA: IEEE and Maple Press,(1988), 681-689.
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weights for nuclear safeguards evaluation, Knowledge-Based Systems, 28 (2012), 19-26.
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(1997), 153-166.
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[14] S. P. Wan, 2-tuple linguistic hybrid arithmetic aggregation operators and application to
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multi-attribute group decision making, Knowledge-Based Systems, 45 (2013), 31-40.
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[15] S. P.Wan, Some hybrid geometric aggregation operators with 2-tuple linguistic Information
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and their applications to multi-attribute group decision making, International Journal of
33
Computational Intelligence Systems, 6(4) (2013), 750-763.
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[16] S. P. Wan and J. Y. Dong. Interval-valued intuitionistic fuzzy mathematical programming
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method for hybrid multi-criteria group decision making with interval-valued intuitionistic
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fuzzy truth degrees, Information Fusion, 26 (2015), 49-65.
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[17] S. P. Wan and D. F. Li. Fuzzy mathematical programming approach to heterogeneous
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multiattribute decision-making with interval-valued intuitionistic fuzzy truth degrees, Information
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Sciences, 325 (2015), 484-503.
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41
valued intuitionistic fuzzy MAGDM with incomplete attribute weight information, Information
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Sciences, 316 (2015), 329-347.
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granularity intuitionistic two semantics, Science and Technology Information, 33 (2009),
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intuitionistic linguistic fuzzy numbers, Control and Decision, 25(10) (2010), 1571-1574.
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and Theory, 38 (2008), 51-61.
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WG and ET-OWG operators with 2-tuple linguistic information, Expert Systems with
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Application, 37(12) (2010), 7895-7900.
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application to multiple attribute group decision making, Computers and Industrial Engineering,
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61(1) (2011), 32-38.
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group decision making with incomplete weight information, Expert Systems with Application,
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38(5) (2011), 7895-7900.
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group decision making under uncertain linguistic environment, Information Sciences, 168
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Information Fusion, 7 (2006), 231-238.
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69
ators to group decision making with uncertain multiplicative linguistic preference relation,
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[30] Z. S. Xu, An interactive approach to multiple attribute group decision making with multi-
72
granular uncertain linguistic information, Group Decision and Negotiation, 18 (2009),
73
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multiplicative linguistic preference relations, International Journal of Uncertainty, Fuzziness
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making, IEEE Trans Syst Man Cybern,18 (1988), 183-190.
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IEEE Transactions on Fuzzy Systems, (1998), 6286-294.
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90
ORIGINAL_ARTICLE
Decision making in medical investigations using new divergence measures for intuitionistic fuzzy sets
In recent times, intuitionistic fuzzy sets introduced by Atanassov has been one of the most powerful and flexible approaches for dealing with complex and uncertain situations of real world. In particular, the concept of divergence between intuitionistic fuzzy sets is important since it has applications in various areas such as image segmentation, decision making, medical diagnosis, pattern recognition and many more. The aim of this paper is to introduce a new divergence measure for Atanassov's intuitionistic fuzzy sets (textit{AIFS)}. The properties of the proposed divergence measure have been studied and the findings are applied in medical diagnosis of some diseases with a common set of symptoms.
http://ijfs.usb.ac.ir/article_2285_c662f28b659682009004d67579186183.pdf
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44
10.22111/ijfs.2016.2285
Fuzzy sets
Intuitionistic fuzzy sets
Divergence measure
Medical diagnosis
A.
Srivastava
true
1
Department of Mathematics, Jaypee Institute of Information Tech-
nology, Noida, Uttar Pradesh
Department of Mathematics, Jaypee Institute of Information Tech-
nology, Noida, Uttar Pradesh
Department of Mathematics, Jaypee Institute of Information Tech-
nology, Noida, Uttar Pradesh
LEAD_AUTHOR
S.
Maheshwari
true
2
Department of Mathematics, Jaypee Institute of Information Tech-
nology, Noida, Uttar Pradesh
Department of Mathematics, Jaypee Institute of Information Tech-
nology, Noida, Uttar Pradesh
Department of Mathematics, Jaypee Institute of Information Tech-
nology, Noida, Uttar Pradesh
AUTHOR
[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.
1
[2] D. Bhandari and N. R. Pal, Some new information measure for fuzzy sets, Information
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Science, 67 (1993), 209-228.
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[3] F. E. Boran and D. Akay, A biparametric similarity measure on intuitionistic fuzzy sets with
4
applications to pattern recognition, Information Sciences, 255(10) (2014) 45-57.
5
[4] T. Chaira and A. K. Ray, Segmentation using fuzzy divergence, Pattern Recognition Letters,
6
24(12) (2003), 1837-1844.
7
[5] S. K. De, R. Biswas and A. R. Roy, An application of intuitionistic fuzzy sets in medical
8
diagnosis, Fuzzy Sets and Systems, 117(2) (2001), 209-213.
9
[6] J. Fan and W. Xie, Distance measure and induced fuzzy entropy, Fuzzy Sets and Systems,
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104(2) (1999), 305-314.
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[7] A. G. Hatzimichailidis, G. A. Papakostas and V. G. Kaburlasos, A novel distance measure
12
of intuitionistic fuzzy sets and its application to pattern recognition problems, International
13
Journal of Intelligent Systems 27(4) (2012), 396-409.
14
[8] W. L. Hung and M. S. Yang, On the J- Divergence of intuitionistic fuzzy sets with its
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applications to pattern recognition, Information sciences, 178(6) (2008), 1641-1650.
16
[9] K. C. Hung, Medical pattern recognition: applying an improved intuitionistics fuzzy cross-
17
entropy approach, Advances in fuzzy Systems Article ID, 863549 (2012).
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[10] M. Junjun, Y. Dengbao and W. Cuicui, A novel cross-entropy and entropy measures of IFSs
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and their applications, Knowledge-Based Systems, 48 (2013), 37-45.
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tion Theory, 37(1) (1991), 145-151.
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[13] S. Montes, I. Couso, P. Gil and C. Bertoluzza, Divergence measure between fuzzy sets, Inter-
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national Journal of Approximate Reasoning, 30(2) (2002), 91-105.
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[14] I. Montes, V. Janis and S. Montes, An axiomatic denition of divergence for intuitionistic
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fuzzy sets, Advances in Intelligent Systems Research, EUSFLAT 2011, Atlantis Press, Aix-Les
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Bains, ISBN 978-90-78677-00-0, (2011), 547-553.
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[15] I. Montes, N. R. Pal, V. Janis and S. Montes, Divergence measures for intuitionistic fuzzy
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sets, IEEE Transactions on Fuzzy Systems, 23 (2015) 444-456.
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[16] G. A. Papakostas, A. G. Hatzimichailidis and V. G. Kaburlasos, Distance and similarity
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measures between intuitionistic fuzzy sets: a comparative analysis from a pattern recognition
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point of view, Pattern Recognition Letters, 34(14) (2013), 1609-1622.
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Letters, 18(5) (1997), 425-432
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diagnosis, Proceedings of the Computational Science ICCS. Springer, Berlin, Germany, 2074,
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(2001), 263-271.
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[19] E. Szmidt and J. Kacprzyk, Intuitionistic fuzzy sets in some medical applications, Pro-
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ceedings of the 7th Fuzzy Days, 2206 Computational intelligence: theory and applications.
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Springer, Berlin, Germany (2001), 148-151.
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[20] E. Szmidt and J. Kacprzyk, A Similarity Measure for Intuitionistic Fuzzy Sets and its Ap-
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plication in Supporting Medical Diagnostic Reasoning, Articial Intelligence and Soft Com-
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puting { ICAISC, 3070 (2004), 388-393.
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[21] K. Vlachos and G. D. Sergiadis, Intuitionistic fuzzy information|applications to pattern
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recognition, Pattern Recognition Letters, 28(2) (2007), 197-206.
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recognition, International Conference on Intelligent Systems and Knowledge Engineering,
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fuzzy environment, Information Fusion, 13(1) (2012), 31-47.
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applications, Information Sciences, 178(21) (2008), 4184-4191.
54
ORIGINAL_ARTICLE
Designing a model of intuitionistic fuzzy VIKOR in multi-attribute group decision-making problems
Multiple attributes group decision making (MAGDM) is regarded as the process of determining the best feasible solution by a group of experts or decision makers according to the attributes that represent different effects. In assessing the performance of each alternative with respect to each attribute and the relative importance of the selected attributes, quantitative/qualitative evaluations are often required to handle uncertainty, imprecise and inadequate information, which are well suited to represent with fuzzy values. This paper develops a VIKOR method based on intuitionistic fuzzy sets with multi-judges and multi-attributes in real-life situations. Intuitionistic fuzzy weighted averaging (IFWA) operator is used to aggregate individual judgments of experts to rate the importance of attributes and alternatives. Then, an intuitionistic ranking index is introduced to obtain a compromise solution to solve MAGDM problems. For application and validation, this paper presents two application examples and solves the practical portfolio selection and material handling selection problems to verify the proposed method. Finally, the intuitionistic fuzzy VIKOR method is compared with the existing intuitionistic fuzzy MAGDM method for two application examples, and their computational results are discussed.
http://ijfs.usb.ac.ir/article_2286_459119c7ea81ef16c1dc2c156e716ea4.pdf
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65
10.22111/ijfs.2016.2286
Intuitionistic fuzzy sets
VIKOR method
Group decision making
Portfolio selection
Material handling selection
Seyed Meysam
Mousavi
true
1
Department of Industrial Engineering, Faculty of Engi-
neering, Shahed University, Tehran, Iran
Department of Industrial Engineering, Faculty of Engi-
neering, Shahed University, Tehran, Iran
Department of Industrial Engineering, Faculty of Engi-
neering, Shahed University, Tehran, Iran
LEAD_AUTHOR
Behnam
Vahdani
b.vahdani@ut.ac.ir
true
2
Faculty of Industrial and Mechanical Engineering, Qazvin Branch,
Islamic Azad University, Qazvin, Iran
Faculty of Industrial and Mechanical Engineering, Qazvin Branch,
Islamic Azad University, Qazvin, Iran
Faculty of Industrial and Mechanical Engineering, Qazvin Branch,
Islamic Azad University, Qazvin, Iran
AUTHOR
Shadan Sadigh
Behzadi
true
3
Department of Mathematics, Qazvin Branch, Islamic Azad
University, Qazvin, Iran
Department of Mathematics, Qazvin Branch, Islamic Azad
University, Qazvin, Iran
Department of Mathematics, Qazvin Branch, Islamic Azad
University, Qazvin, Iran
AUTHOR
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ing with incomplete fuzzy linguistic preference relations, International Journal of Intelligent
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Systems., 24(2) (2009), 201-222.
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[2] M. Amiri-Aref, N. Javadian N, A new fuzzy TOPSIS method for material handling system
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selection problems, Proceedings of the 8th WSEAS International Conference on Software
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Engineering, Parallel and Distributed Systems., (2009), 169-174.
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Systems., 61 (1994), 137-142.
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9(1) (2003), 71-75.
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making for supplier selection with TOPSIS method, Expert Systems with Applications., 36
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(2009), 11363-11368.
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fuzzy sets, Fuzzy Sets and Systems., 78 (1996), 305-316.
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multi-criteria decision approach, Mathematics and Computers in Simulation., 77 (2008),
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strategic decision of fuzzy VIKOR, International Journal of Production Economics., 120
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tions, Springer-Verlag, Berlin., 1992.
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exible-joint robot arm, Information Science., 145 (2002), 13-43.
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with words, IEEE Transactions on Fuzzy Systems., 8 (2000), 746- 752.
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International Journal of Approximate Reasoning., 46 (2007), 120-136.
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fuzzy sets, Journal of Computer and System Sciences., 73 (2007), 84-88.
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making approach for new product selection in a fuzzy environment, Arabian Journal for
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multi-stage decision making process for multiple attributes analysis under an interval-valued
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analysis of VIKOR and TOPSIS, European Journal of Operational Research., 156 (2004),
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[40] S. Opricovic and G-H. Tzeng, Extended VIKOR method in comparison with outranking meth-
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ods, European Journal of Operational Research., 178 (2007), 514-529.
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lection with VIKOR under fuzzy environment, Expert Systems with Applications., 37 (2010),
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VIKOR method, International Journal of Advanced Manufacturing Technology., 31 (2007),
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1049-1057.
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location in Taipei, International Journal of Hospitality Management., 21 (2002), 171-187.
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elimination and choice translating reality method for multiple criteria group decision-making
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computing with words, IEEE Transactions on Fuzzy Systems., 14 (2010), 435-445.
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our country with interval intuitionistic fuzzy information, Advances in information sciences
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linguistic preference relations, Information Sciences., 166 (2004), 19-30.
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sets, International Journal of General Systems., 35 (2006), 417-433.
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making method for personnel selection, Expert Systems with Applications., 38 (2011), 11401-
134
ORIGINAL_ARTICLE
Piecewise cubic interpolation of fuzzy data based on B-spline basis functions
In this paper fuzzy piecewise cubic interpolation is constructed for fuzzy data based on B-spline basis functions. We add two new additional conditions which guarantee uniqueness of fuzzy B-spline interpolation.Other conditions are imposed on the interpolation data to guarantee that the interpolation function to be a well-defined fuzzy function. Finally some examples are given to illustrate the proposed method.
http://ijfs.usb.ac.ir/article_2287_4c7e790202a7257a98a74eed659ffe8c.pdf
2016-02-28T11:23:20
2018-02-25T11:23:20
67
76
10.22111/ijfs.2016.2287
Fuzzy number
Fuzzy interpolation
B-spline basis functions
Masoumeh
Zeinali
zeinali@tabrizu.ac.ir
true
1
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
AUTHOR
Sedaghat
Shahmorad
shahmorad@tabrizu.ac.ir
true
2
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
LEAD_AUTHOR
Kamal
Mirnia
mirnia-kam@tabrizu.ac.ir
true
3
Faculty of mathematical sciences, University of Tabriz, Tabriz, Iran
Faculty of mathematical sciences, University of Tabriz, Tabriz, Iran
Faculty of mathematical sciences, University of Tabriz, Tabriz, Iran
AUTHOR
[1] A. M. Anile, B. Falcidieno, G. Gallo, M. Spagnuolo and S. Spinello, Modeling uncertain data
1
with fuzzy B-spline, Fuzzy Sets and Systems, 113 (2000), 397-410.
2
[2] B. Bede and S. G. Gal, Generalizations of the dierentiability of fuzzy-number-valued func-
3
tions with applications to fuzzy dierential equations, Fuzzy Sets and Systems, 151 (2005),
4
[3] P. Blaga and B. Bede, Approximation by fuzzy B-spline series, J. Appl. Math. & Computing,
5
20 (2006), 157169.
6
[4] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986),
7
[5] O. Kaleva, Interpolation of fuzzy data, Fuzzy Sets and Systems, 61 (1994), 63-70.
8
[6] W. A. Lodwick and J. Santos, Constructing consistenct fuzzy surfaces from fuzzy data, Fuzzy
9
Sets and Systems, 135 (2003), 259-277.
10
[7] R. Lowen, A fuzzy Lagrange interpolation theorem, Fuzzy Sets and Systems, 34 (1990), 33-38.
11
[8] P. M. Prenter, Spline and variational methods, A Wiley-Interscience publication, 1975.
12
[9] C. Wu and Z. Gong, On Henstock integral of fuzzy-number-valued functions I, Fuzzy Sets
13
and Systems, 120 (2001), 523-532.
14
[10] M. Zeinali, S. Shahmorad and K. Mirnia, Hermite and piecewise cubic Hermite interpolation
15
of fuzzy data, Journal of Intelligent & Fuzzy Systems, 26 (2014), 2889-2898.
16
ORIGINAL_ARTICLE
Further results on $L$-ordered fuzzifying convergence spaces
In this paper, it is shown that the category of $L$-ordered fuzzifying convergence spaces contains the category of pretopological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory and the latter contains the category of topological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory. Also, it is proved that the category of $L$-ordered fuzzifying convergence spaces can be embedded in the category of stratified $L$-ordered convergence spaces as a coreflective subcategory.
http://ijfs.usb.ac.ir/article_2289_7edf80dce877328f54e067a17901d97c.pdf
2016-02-28T11:23:20
2018-02-25T11:23:20
77
92
10.22111/ijfs.2016.2289
$L$-ordered fuzzifying convergence space
Stratified $L$-ordered convergence space
$L$-filter
Category theory
Bin
Pang
pangbin1205@163.com
true
1
Shenzhen Graduate School, Harbin Institute of Technology, 518055 Shen-
zhen, P.R. China
Shenzhen Graduate School, Harbin Institute of Technology, 518055 Shen-
zhen, P.R. China
Shenzhen Graduate School, Harbin Institute of Technology, 518055 Shen-
zhen, P.R. China
AUTHOR
Yi
Zhao
zhaoyisz420@sohu.com
true
2
Shenzhen Graduate School, Harbin Institute of Technology, 518055 Shen-
zhen, P.R. China
Shenzhen Graduate School, Harbin Institute of Technology, 518055 Shen-
zhen, P.R. China
Shenzhen Graduate School, Harbin Institute of Technology, 518055 Shen-
zhen, P.R. China
LEAD_AUTHOR
[1] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and concrete categories, Wiley, New
1
York, 1990.
2
[2] J. M. Fang, Stratied L-ordered convergence structures, Fuzzy Sets Syst., 161 (2010), 2130{
3
[3] J. M. Fang, Relationships between L-ordered convergence structures and strong L-topologies,
4
Fuzzy Sets Syst., 161(22) (2010), 2923{2944.
5
[4] J. Gutierrez Garca, I. Mardones Perez and M. H. Burton, The relationship between various
6
lter notions on a GL-Monoid, J. Math. Anal. Appl., 230(1999), 291{302.
7
[5] U. Hohle and A. P. Sostak, Axiomatic foudations of xed-basis fuzzy topology, in: U. Hohle,
8
S.E. Rodabaugh(Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory,
9
Handbook Series, vol.3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, 123{
10
[6] U. Hohle, Characterization of L-topologies by L-valued neighborhoods, Chapter 5, In [5],
11
[7] U. Hohle, Many valued topology and its applications, Kluwer Academic Publishers, Boston,
12
Dordrecht, London, 2001.
13
[8] G. Jager, A category of L-fuzzy convergence spaces, Quaest. Math., 24 (2001), 501{517.
14
[9] G. Jager, Subcategories of lattice-valued convergence spaces, Fuzzy Sets Syst., 156 (2005),
15
[10] G. Jager, Pretopological and topological lattice-valued convergence spaces, Fuzzy Sets Syst.,
16
158 (2007), 424{435.
17
[11] B. Y. Lee, J. H. Park and B. H. Park, Fuzzy convergence structures, Fuzzy Sets Syst., 56
18
(1993), 309{315.
19
[12] L. Q. Li and Q. Jin, On stratied L-convergence spaces: Pretopological axioms and diagonal
20
axioms, Fuzzy Sets Syst., 204 (2012), 40{52.
21
[13] E. Lowen, R. Lowen and P. Wuyts, The categorical topological approach to fuzzy topology
22
and fuzzy convergence, Fuzzy Sets Syst., 40 (1991), 347{373.
23
[14] K. C. Min, Fuzzy limit spaces, Fuzzy Sets Syst., 32 (1989), 343{357.
24
[15] B. Pang and J. M. Fang, L-fuzzy Q-convergence structures, Fuzzy Sets Syst., 182 (2011),
25
[16] B. Pang, On (L;M)-fuzzy convergence spaces, Fuzzy Sets Syst., 238 (2014), 46{70.
26
[17] B. Pang, Enriched (L;M)-fuzzy convergence spaces, J. Intell. Fuzzy Syst., 27(1) (2014),
27
[18] B. Pang and F. G. Shi Degrees of compactness of (L;M)-fuzzy convergence spaces and its
28
applications, J. Intell. Fuzzy Syst., 251 (2014), 1{22.
29
[19] W. C. Wu and J. M. Fang, L-ordered fuzzifying convergence spaces, Iranian Journal of Fuzzy
30
Systems, 9(2) (2012), 147{161.
31
[20] L. S. Xu, Characterizations of fuzzifying topologies by some limit structures, Fuzzy Sets Syst.,
32
123 (2001), 169{176.
33
[21] W. Yao, On many-valued stratied L-fuzzy convergence spaces, Fuzzy Sets Syst., 159 (2008),
34
2503{2519.
35
[22] W. Yao, On L-fuzzifying convergence spaces, Iranian Journal of Fuzzy Systems, 6(1) (2009),
36
ORIGINAL_ARTICLE
On The Relationships Between Types of $L$-convergence Spaces
This paper focuses on the relationships between stratified $L$-conver-gence spaces, stratified strong $L$-convergence spaces and stratifiedlevelwise $L$-convergence spaces. It has been known that: (1) astratified $L$-convergence space is precisely a left-continuousstratified levelwise $L$-convergence space; and (2) a stratifiedstrong $L$-convergence space is naturally a stratified $L$-convergence space, but the converse is not true generally.In this paper, a strong left-continuity condition for stratified levelwise $L$-convergence space is given. It is proved that a stratified strong $L$-convergence space is precisely a strongly left-continuous stratifiedlevelwise $L$-convergence space. Then a sufficient and necessary condition for a stratified $L$-convergence space to be a stratified strong $L$-convergence space is presented.
http://ijfs.usb.ac.ir/article_2290_71de97e8ff22e4ed37bf0b947a19e70a.pdf
2016-02-28T11:23:20
2018-02-25T11:23:20
93
103
10.22111/ijfs.2016.2290
$L$-topology
Stratified $L$-filter
Stratified $L$-convergence space
Qiu
Jin
jinqiu79@126.com
true
1
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
AUTHOR
Lingqiang
Li
lilingqiang@126.com
true
2
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
LEAD_AUTHOR
Guangwu
Meng
true
3
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
AUTHOR
[1] R. Belohlavek, Fuzzy relational systems: Foundations and Principles, New York: Kluwer
1
Academic Publishers, (2002), 75-212.
2
[2] J. M. Fang, Stratied L-ordered convergence structures, Fuzzy Sets and Systems, 161 (2010),
3
2130{2149.
4
[3] J. M. Fang, Relationships between L-ordered convergence structures and strong L-topologies,
5
Fuzzy Sets and Systems, 161 (2010), 2923{2944.
6
[4] J. M. Fang, Lattice-valued semiuniform convergence spaces, Fuzzy Sets and Systems, 195
7
(2012), 33{57.
8
[5] J. M. Fang, Stratied L-ordered quasiuniform limit spaces, Fuzzy Sets and Systems, 227
9
(2013), 51{73.
10
[6] P. V. Flores, R. N. Mohapatra and G. Richardson, Lattice-valued spaces: Fuzzy convergence,
11
Fuzzy Sets and Systems, 157 (2006), 2706{2714.
12
[7] U. Hohle, Commutative, residuated l-monoids, In: U. Hohle, E.P. Klement (Eds.), Nonclassical
13
Logics and Their Applications to Fuzzy Subsets: A Handbook of the Mathematical
14
Foundations of Fuzzy Set Theory, Dordrecht: Kluwer Academic Publishers, (1995), 53{105.
15
[8] U. Hohle and A. Sostak, Axiomatic foundations of xed-basis fuzzy topology, In: U. Hohle,
16
S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory,
17
The Handbooks of Fuzzy Sets Series, Vol.3, Boston, Dordrecht, London: Kluwer Academic
18
Publishers, (1999), 123{273.
19
[9] G. Jager, A category of L-fuzzy convergence spaces, Quaestiones Mathematicae, 24 (2001),
20
[10] G. Jager, Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems, 156
21
(2005), 1{24.
22
[11] G. Jager, Fischer's diagonal condition for lattice-valued convergence spaces, Quaestiones
23
Mathematicae, 31 (2008), 11{25.
24
[12] G. Jager, Gahler's neighbourhood condition for lattice-valued convergence spaces, Fuzzy Sets
25
and Systems, 204 (2012), 27{39.
26
[13] G. Jager, Diagonal conditions for lattice-valued uniform convergence spaces, Fuzzy Sets and
27
Systems, 210 (2013), 39{53.
28
[14] G. Jager, Stratied LMN-convergence tower spaces, Fuzzy Sets and Systems, 282 (2016),
29
[15] L. Li and Q. Li, A new regularity (p-regularity) of stratied L-generalized convergence spaces,
30
Journal of Computational Analysis and Applications, 20(2) (2016), 307-318.
31
[16] L. Li and Q. Jin, On adjunctions between Lim, SL-Top, and SL-Lim, Fuzzy Sets and Systems,
32
182 (2011), 66{78.
33
[17] L. Li and Q. Jin, On stratied L-convergence spaces: Pretopological axioms and diagonal
34
axioms, Fuzzy Sets and Systems, 204 (2012), 40{52.
35
[18] L. Li and Q. Jin, p-topologicalness and p-regularity for lattice-valued convergence spaces,
36
Fuzzy Sets and Systems, 238 (2014), 26{45.
37
[19] L. Li and Q. Jin, lattice-valued convergence spaces: weaker regularity and p-regularity, Abstract
38
and Applied Analysis, Volume 2014, Article ID 328153, 11 pages.
39
[20] L. Li, Q. Jin and K. Hu, On stratied L-convergence spaces: Fischer's diagonal axiom, Fuzzy
40
Sets and Systems, 267 (2015), 31{40.
41
[21] L. Li, Q. Jin, G. Meng and K. Hu, The lower and upper p-topological (p-regular) modications
42
for lattice-valued convergence spaces, Fuzzy Sets and Systems, 282 (2016), 47{61.
43
[22] D. Orpen and G. Jager, Lattice-valued convergence spaces: extending the lattice context,
44
Fuzzy Sets and Systems, 190 (2012), 1{20.
45
[23] B. Pang and F. Shi, Degrees of compactness in (L;M)-fuzzy convergence spaces, Fuzzy Sets
46
and Systems, 251 (2014), 1{22.
47
[24] G. D. Richardson and D. C. Kent, Probabilistic convergence spaces, Journal of the Australian
48
Mathematical Society, 61 (1996), 400{420.
49
[25] W. Yao, On many-valued stratied L-fuzzy convergence spaces, Fuzzy Sets and Systems, 159
50
(2008), 2503{2519.
51
[26] W. Yao, Quantitative domains via fuzzy sets: Part I: continuity of fuzzy directed complete
52
posets, Fuzzy Sets and Systems, 161 (2010), 973{987.
53
[27] W. Yao and F. Shi, Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on
54
fuzzy directed-complete posets, Fuzzy Sets and Systems, 173 (2011), 60{80.
55
[28] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems,
56
158 (2007), 349{366.
57
[29] Q. Zhang, W. Xie and L. Fan, Fuzzy complete lattices, Fuzzy Sets and Systems, 160 (2009),
58
2275{2291.
59
ORIGINAL_ARTICLE
Correspondence between probabilistic norms and fuzzy norms
In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.
http://ijfs.usb.ac.ir/article_2291_c21f5bbd1e1e97f9ce8d83b9838715be.pdf
2016-02-28T11:23:20
2018-02-25T11:23:20
105
114
10.22111/ijfs.2016.2291
Probabilistic norm
Fuzzy norm
Hua-Peng
Zhang
huapengzhang@163.com
true
1
School of Science, Nanjing University of Posts and Telecommuni-
cations, Nanjing 210023, China
School of Science, Nanjing University of Posts and Telecommuni-
cations, Nanjing 210023, China
School of Science, Nanjing University of Posts and Telecommuni-
cations, Nanjing 210023, China
LEAD_AUTHOR
[1] C. Alegre and S. Romaguera, Characterizations of metrizable topological vector spaces and
1
their asymmetric generalizations in terms of fuzzy (quasi-)norms, Fuzzy Sets and Systems,
2
161 (2010), 2181{2192.
3
[2] C. Alsina, M. J. Frank and B. Schweizer, Associative Functions: Triangular Norms and
4
Copulas, World Scientic Publishing, Singapore, 2006.
5
[3] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math.,
6
11(3) (2003), 687{705.
7
[4] T. Bag and S. K. Samanta, A comparative study of fuzzy norms on a linear space, Fuzzy
8
Sets and Systems, 159 (2008), 670{684.
9
[5] S. C. Cheng and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces,
10
Bull. Calcutta Math. Soc., 86 (1994), 429{436.
11
[6] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48 (1992),
12
[7] O. Hadzic and E. Pap, Fixed point theory in probabilistic metric spaces, Kluwer Academic
13
Publishers, Dordrecht, 2001.
14
[8] U. Hohle, Minkowski functionals of L-fuzzy sets, in: P.P. Wang, S.K. Chang (Eds.), Fuzzy
15
sets: theory and applications to policy analysis and information systems, Plenum Press, New
16
York, (1980), 1324.
17
[9] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),
18
[10] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12 (1984), 143{
19
[11] A. K. Katsaras, Linear fuzzy neighborhood spaces, Fuzzy Sets and Systems, 16 (1985), 25{40.
20
[12] A. K. Katsaras, Locally convex topologies induced by fuzzy norms, Global Journal of Mathematical
21
Analysis, 1(3) (2013), 83{96.
22
[13] E. P. Klement, R. Mesiar and E. Pap, Triangular norms, Kluwer Academic Publishers,
23
Dordrecht, 2000.
24
[14] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11
25
(1975), 336{344.
26
[15] B. Lafuerza-Guillen and P. K. Harikrishnan, Probabilistic normed spaces, World Scientic
27
Publishing, Singapore, 2014.
28
[16] M. Ma, A comparison between two denitions of fuzzy normed spaces, J. Harbin Inst. Technology
29
Suppl. Math., (in Chinese), (1985), 47{49.
30
[17] S. Nadaban and I. Dzitac, Atomic decompositions of fuzzy normed linear spaces for wavelet
31
applications, Informatica, 25(4) (2014), 643{662.
32
[18] R. Saadati and S. M. Vaezpour, Some results on fuzzy Banach spaces, J. Appl. Math. &
33
Computing, 17(1-2) (2005), 475{484.
34
[19] B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland series in Probability
35
and Applied Mathematics, North-Holland, New York, 1983.
36
[20] C. Sempi, A short and partial history of probabilistic normed spaces, Mediterr. J. Math., 3
37
(2006), 283{300.
38
[21] C. X. Wu and J. X. Fang, Fuzzy generalization of Kolmogoro's theorem, J. Harbin Inst.
39
Technology, (in Chinese), 1 (1984), 1{7.
40
[22] C. X. Wu and M. Ma, Fuzzy norms, probabilistic norms and fuzzy metrics, Fuzzy Sets and
41
Systems, 36 (1990), 137{144.
42
[23] C. H. Yan and J. X. Fang, Generalization of Kolmogoro's theorem to L-topological vector
43
spaces, Fuzzy Sets and Systems, 125 (2002), 177{183.
44
ORIGINAL_ARTICLE
$L$-fuzzy approximation spaces and $L$-fuzzy topological spaces
The $L$-fuzzy approximation operator associated with an $L$-fuzzy approximation space $(X,R)$ turns out to be a saturated $L$-fuzzy closure (interior) operator on a set $X$ precisely when the relation $R$ is reflexive and transitive. We investigate the relations between $L$-fuzzy approximation spaces and $L$-(fuzzy) topological spaces.
http://ijfs.usb.ac.ir/article_2292_6313286d0ad372c6aaa06e50234658ed.pdf
2016-02-28T11:23:20
2018-02-25T11:23:20
115
129
10.22111/ijfs.2016.2292
Complete residuated lattice
$L$-fuzzy approximation spaces
$L$-fuzzy topology
Continuity
A. A.
Ramadan
true
1
Department of Mathematics, Faculty of Science, Beni-Suef Univer-
sity, Beni-Suef, Egypt
Department of Mathematics, Faculty of Science, Beni-Suef Univer-
sity, Beni-Suef, Egypt
Department of Mathematics, Faculty of Science, Beni-Suef Univer-
sity, Beni-Suef, Egypt
LEAD_AUTHOR
E. H.
Elkordy
true
2
Department of Mathematics, Faculty of Science, Beni-Suef Univer-
sity, Beni-Suef, Egypt
Department of Mathematics, Faculty of Science, Beni-Suef Univer-
sity, Beni-Suef, Egypt
Department of Mathematics, Faculty of Science, Beni-Suef Univer-
sity, Beni-Suef, Egypt
AUTHOR
M.
El-Dardery
true
3
Department of Mathematics, Faculty of Science, Fayoum University,
Fayoum, Egypt
Department of Mathematics, Faculty of Science, Fayoum University,
Fayoum, Egypt
Department of Mathematics, Faculty of Science, Fayoum University,
Fayoum, Egypt
AUTHOR
[1] R. Belohlavek, Fuzzy relational systems: foundations and principles, Kluwer Academic/
1
Plenum Press, New York (2002).
2
[2] K. Blount and T. Tsinakis, The structure of residuated lattices, Int. J. Algebra and Computation,
3
13(4) (2004), 473{461.
4
[3] D. Boixader, J. Jacas and J. Recasens, Upper and lower approximations of fuzzy sets, Int.
5
Jour. of Gen. Sys., 29 (2000), 555{568.
6
[4] X. Chen and Q. Li, Construction of rough approximations in fuzzy setting, Fuzzy Sets and
7
Systems, 158 (2007), 2641{2653.
8
[5] M. Chuchro, On rough sets in topological Boolean algebra. In: Ziarko, W.(ed.): Rough Sets,
9
Fuzzy Sets and Knowledge Discovery, Springer-Verlage, New York, (1994), 157{160.
10
[6] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst., 17(2-3)
11
(1990), 191{208.
12
[7] J. Fang, I-fuzzy Alexadrov topologies and specialization orders, Fuzzy Sets and Systems, 158
13
(2007), 2359{2374.
14
[8] P. Hajek, Metamathematics of fuzzy logic, Kluwer, Dordrecht (1998).
15
[9] U. Hohle and A. P. Sostak, Axiomatic foundations of xed-basis fuzzy topology, In: Hohle,
16
S. E. Rodabaugh (Eds), Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory,
17
The Handbooks of Fuzzy Sets Series, Chapter 3, Kluwer Academic Publisher, Dordrechet
18
(1999), 123{272.
19
[10] Y. C. Kim and Y. S. Kim, (L,)-approximation spaces and (L,)-fuzzy quasi-uniform spaces,
20
Information Sciences, 179 (2009), 2028{2048.
21
[11] H. Lai and D. Zhang, Fuzzy pre order and fuzzy topology, Fuzzy Sets and Systems, 157
22
(2006), 1865{1885.
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56
ORIGINAL_ARTICLE
Commutative pseudo BE-algebras
The aim of this paper is to introduce the notion of commutative pseudo BE-algebras and investigate their properties.We generalize some results proved by A. Walendziak for the case of commutative BE-algebras.We prove that the class of commutative pseudo BE-algebras is equivalent to the class of commutative pseudo BCK-algebras. Based on this result, all results holding for commutative pseudo BCK-algebras also hold for commutative pseudo BE-algebras. For example, any finite commutative pseudo BE-algebra is a BE-algebra, and any commutative pseudo BE-algebra is a join-semilattice. Moreover, if a commutative pseudo BE-algebra is a meet-semilattice, then it is a distributive lattice. We define the pointed pseudo-BE algebras, and introduce and study the relative negations on pointed pseudo BE-algebras. Based on the relative negations we construct two closure operators on a pseudo BE-algebra.We also define relative involutive pseudo BE-algebras, we investigate their properties and prove equivalent conditions for a relative involutive pseudo BE-algebra.We define the relative Glivenko property for a relative good pseudo BE-algebra and show that any relativeinvolutive pseudo BE-algebra has the relative Glivenko property.
http://ijfs.usb.ac.ir/article_2293_89fc397124e86bd8d36d0f72b59437ba.pdf
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10.22111/ijfs.2016.2293
Pseudo BE-algebra
Pseudo BCK-algebra
Commutative pseudo BCK-algebra
Commutative pseudo BE-algebra
Pointed pseudo BE-algebra
Relative involutive pseudo BE-algebra
Relative Glivenko property
L. C.
Ciungu
lcciungu@yahoo.com
true
1
Department of Mathematics, University of Iowa, 14 MacLean Hall,
Iowa City, Iowa 52242-1419, Usa
Department of Mathematics, University of Iowa, 14 MacLean Hall,
Iowa City, Iowa 52242-1419, Usa
Department of Mathematics, University of Iowa, 14 MacLean Hall,
Iowa City, Iowa 52242-1419, Usa
LEAD_AUTHOR
[1] S. S. Ahn and K. S. So, On ideals and upper sets in BE-algebras, Scientiae Mathematicae
1
Japonicae, 68(2) (2008), 279-285.
2
[2] S. S. Ahn, Y. H. Kim and J. M. Ko, Filters in commutative BE-algebras, Communications
3
of the Korean Mathematical Society, 27(2) (2012), 233-242.
4
[3] A. Borumand Saeid, Smarandache BE-algebras, Education Publisher, Columbus, Ohio, USA,
5
[4] R. A. Borzooei, A. Borumand Saeid, A. Rezaei, A. Radfar and R. Ameri, On pseudo BE-
6
algebras, Discussiones Mathematicae General Algebra and Applicationes, 33(1) (2013), 95-
7
[5] R. A. Borzooei, A. Borumand Saeid and R. Ameri, States on BE-algebras, Kochi Journal of
8
Mathematics, 9(1) (2014), 27-42.
9
[6] R. A. Borzooei, A. Borumand Saeid, A. Rezaei, A. Radfar and R. Ameri, Distributive pseudo
10
BE{algebras, Fasciculi Mathematici, 54(1) (2015), 21-39.
11
[7] R. Cignoli and A. Torrens, Glivenko like theorems in natural expansions of BCK-logic, Mathematical
12
Logic Quaterly, 50(2) (2004), 111-125.
13
[8] R. Cignoli and A. Torrens, Free Algebras in Varieties of Glivenko MTL-algebras Satisfying
14
the Equation 2(x2) = (2x)2, Studia Logica, 83(1-3) (2006), 157-181.
15
[9] Z. Ciloglu and Y. Ceven, Commutative and bounded BE-algebras, Hindawi Publishing Corporation,
16
2013(1) (2013), Article ID 473714.
17
[10] L. C. Ciungu and A. Dvurecenskij, Measures, states and de Finetti maps on pseudo-BCK
18
algebras, Fuzzy Sets and Systems, 161(22) (2010), 2870-2896.
19
[11] L. C. Ciungu, G. Georgescu and C. Muresan, Generalized Bosbach states: part I, Archive for
20
Mathematical Logic, 52(3-4) (2013), 335-376.
21
[12] L. C. Ciungu and J. Kuhr, New probabilistic model for pseudo-BCK algebras and pseudo-
22
hoops, Journal of Multiple-Valued Logic and Soft Computing, 20(3-4) (2013), 373-400.
23
[13] L. C. Ciungu, Non-commutative multiple-valued logic algebras, Springer, Cham, Heidelberg,
24
New York, Dordrecht, London, 2014.
25
[14] L. C. Ciungu, Relative negations in non-commutative fuzzy structures, Soft Computing,
26
18(1) (2014), 15-33.
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10.1007/s00500-015-1888-x, (2015).
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and Soft Computing, 12(1-2) (2006), 71-130.
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22(2) (2011), 115-127.
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[34] A. Rezaei, A. Borumand Saeid, A. Radfar and R. A. Borzooei, Congruence relations on
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pseudo BE-algebras, Annnals of the University of Craiova, Mathematics and Computer Sciences
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Series, 41(2) (2014), 166-176.
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[35] A. Rezaei, L. C. Ciungu and A. Borumand Saeid, States on pseudo BE-algebras, submitted.
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[36] A. Walendziak, On commutative BE-algebras, Scientiae Mathematicae Japonicae, 69(2)
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(2009), 281-284.
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[37] A. Walendziak, On normal lters and congruence relations in BE-algebras, Commentationes
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Mathematicae, 52(2) (2012), 199-205.
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residuated lattices, Fuzzy Sets and Systems, 187(1) (2012), 33-57.
69
ORIGINAL_ARTICLE
Semi-G-filters, Stonean filters, MTL-filters, divisible filters, BL-filters and regular filters in residuated lattices
At present, the filter theory of $BL$textit{-}algebras has been widelystudied, and some important results have been published (see for examplecite{4}, cite{5}, cite{xi}, cite{6}, cite{7}). In other works such ascite{BP}, cite{vii}, cite{xiii}, cite{xvi} a study of a filter theory inthe more general setting of residuated lattices is done, generalizing thatfor $BL$textit{-}algebras. Note that filters are also characterized byvarious types of fuzzy sets. Most of such characterizations is trivial butsome are nontrivial, for example characterizations obtained in cite{xm}.Both situation have revealed a rich range of classes of filters: Boolean,implicative, Heyting, positive implicative, fantastic (or MV-filter), etc.In this paper we work in the general cases of residuated lattices and put inevidence new types of filters in a residuated lattice (in the spirit of cite{mvl}): semi-G-filterstextit{, }Stonean filters, divisible filters,BL-filters and regular filters.
http://ijfs.usb.ac.ir/article_2294_d8dfab9476be9a9b5c4b01faa4c2942b.pdf
2016-02-28T11:23:20
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160
10.22111/ijfs.2016.2294
Residuated lattice
Boolean algebra
BL-algebra
MV-algebra
MTL-algebra
Divisible residuated lattice
Regular residuated lattice
Deductive system
Filter
Boolean filter
MTL-filter
Divisible filter
BL-filter
MV-filter
Semi-G-filter
Stonean filter
Regular filter
D.
Busneag
true
1
Department of Mathematics, Faculty of Mathematics and Natural Sci-
ences, University of Craiova, Craiova, Romania
Department of Mathematics, Faculty of Mathematics and Natural Sci-
ences, University of Craiova, Craiova, Romania
Department of Mathematics, Faculty of Mathematics and Natural Sci-
ences, University of Craiova, Craiova, Romania
LEAD_AUTHOR
D.
Piciu
true
2
Department of Mathematics, Faculty of Mathematics and Natural Sciences,
University of Craiova, Craiova, Romania
Department of Mathematics, Faculty of Mathematics and Natural Sciences,
University of Craiova, Craiova, Romania
Department of Mathematics, Faculty of Mathematics and Natural Sciences,
University of Craiova, Craiova, Romania
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ORIGINAL_ARTICLE
Persian-translation vol. 13, no. 1, February 2016
http://ijfs.usb.ac.ir/article_2632_c03568e8d3443db7f6176c58d18a986b.pdf
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172
10.22111/ijfs.2016.2632