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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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
[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1)(1986), 87-96.
1
[2] K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and
2
Systems, 31 (3) (1989), 343-349.
3
[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.
9
[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.
16
[9] J. M. Merigo and A. M. Gil-Lafuente, Decision making techniques in business and econom-
17
ics based on the OWA operator, SORT C Statistics and Operations Research Transactions,
18
36(1) (2012), 81-102.
19
[10] J. M. Merigo, A. M. Gil-Lafuente, L. G. Zhou and H. Y. Chen, Induced and linguistic
20
generalized aggregation operators and their application in linguistic group decision making,
21
Group Decision and Negotiation, 21 (2012), 531-549.
22
[11] M. OHagan, Aggregating template rule antecedents in real-time expert systems with fuzzy
23
set logic. In: Proc 22nd Annual IEEE Asilomar Conference on Signals, Systems and Computers.
24
Pacic Grove, CA: IEEE and Maple Press,(1988), 681-689.
25
[12] Z. Pei, D. Ruan, J. Liu and Y. Xu, A linguistic aggregation operator with three kinds of
26
weights for nuclear safeguards evaluation, Knowledge-Based Systems, 28 (2012), 19-26.
27
[13] V. Torra, The weighted OWA operator, International Journal of Intelligent Systems, 12
28
(1997), 153-166.
29
[14] S. P. Wan, 2-tuple linguistic hybrid arithmetic aggregation operators and application to
30
multi-attribute group decision making, Knowledge-Based Systems, 45 (2013), 31-40.
31
[15] S. P.Wan, Some hybrid geometric aggregation operators with 2-tuple linguistic Information
32
and their applications to multi-attribute group decision making, International Journal of
33
Computational Intelligence Systems, 6(4) (2013), 750-763.
34
[16] S. P. Wan and J. Y. Dong. Interval-valued intuitionistic fuzzy mathematical programming
35
method for hybrid multi-criteria group decision making with interval-valued intuitionistic
36
fuzzy truth degrees, Information Fusion, 26 (2015), 49-65.
37
[17] S. P. Wan and D. F. Li. Fuzzy mathematical programming approach to heterogeneous
38
multiattribute decision-making with interval-valued intuitionistic fuzzy truth degrees, Information
39
Sciences, 325 (2015), 484-503.
40
[18] S. P. Wan, G. L Xu, F. Wang and J. Y. Dong, A new method for Atanassov's interval-
41
valued intuitionistic fuzzy MAGDM with incomplete attribute weight information, Information
42
Sciences, 316 (2015), 329-347.
43
[19] J. Q. Wang and J. J. Li, The multi-criteria group decision making method based on multi-
44
granularity intuitionistic two semantics, Science and Technology Information, 33 (2009),
45
[20] J. Q.Wang and H. B. Li, Multi-criteria decision making based on aggregation operators for
46
intuitionistic linguistic fuzzy numbers, Control and Decision, 25(10) (2010), 1571-1574.
47
[21] Y. Wang and Z. S. Xu, A new method of giving OWA weights, Mathematics in Practice
48
and Theory, 38 (2008), 51-61.
49
[22] G. W. Wei, A method for multiple attribute group decision making based on the ET-
50
WG and ET-OWG operators with 2-tuple linguistic information, Expert Systems with
51
Application, 37(12) (2010), 7895-7900.
52
[23] G. W. Wei, Some generalized aggregating operators with linguistic information and their
53
application to multiple attribute group decision making, Computers and Industrial Engineering,
54
61(1) (2011), 32-38.
55
[24] G. W. Wei, Grey relational analysis method for 2-tuple linguistic multiple attribute
56
group decision making with incomplete weight information, Expert Systems with Application,
57
38(5) (2011), 7895-7900.
58
[25] G. W. Wei and X. F. Zhao, Some dependent aggregation operators with 2-tuple linguistic
59
information and their application to multiple attribute group decision making, Expert
60
Systems with Applications, 39 (2012), 5881-5886.
61
[26] Z. S. Xu, Uncertain linguistic aggregation operators based approach to multiple attribute
62
group decision making under uncertain linguistic environment, Information Sciences, 168
63
(2004), 171-184.
64
[27] Z. S. Xu, An overview of methods for determining OWA weights, International Journal of
65
Intelligent Systems,20(8) (2005), 843-865.
66
[28] Z. S. Xu,Induced uncertain linguistic OWA operators applied to group decision making,
67
Information Fusion, 7 (2006), 231-238.
68
[29] Z. S. Xu,An approach based on the uncertain LOWG and induced uncertain LOWG oper-
69
ators to group decision making with uncertain multiplicative linguistic preference relation,
70
Decision Support Systems, 41 (2006), 488-499.
71
[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
[31] Y. J. Xu and H. M. Wang, Power geometric operators for group decision making under
74
multiplicative linguistic preference relations, International Journal of Uncertainty, Fuzziness
75
and Knowledge-Based Systems, 20(1) (2012), 139-159.
76
[32] R. R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision
77
making, IEEE Trans Syst Man Cybern,18 (1988), 183-190.
78
[33] R. R. Yager, Including importances in OWA aggregation using fuzzy systems modeling,
79
IEEE Transactions on Fuzzy Systems, (1998), 6286-294.
80
[34] R. R. Yager, K. J. Engemann and D. P. Filev, On the concept of immediate probabilities,
81
International Journal of Intelligent Systems 10 (1995), 373-397.
82
[35] W. E. Yang, J. Q.Wang and X. F.Wang, An outranking method for multi-criteria decision
83
making with duplex linguistic information, Fuzzy Sets and Systems, 198 (2012), 20-33.
84
[36] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-356.
85
[37] L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning
86
Part I, Information Sciences, 8(3) (1975), 199-249.
87
[38] S. Z. Zeng and Balezentis, C. H. Zhang, A method based on OWA operator and distance
88
measures for multiple attribute decision making with 2-Tuple linguistic information, Informatica,
89
23(4) (2012), 665-681.
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
2016-02-28T11:23:20
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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
2
Science, 67 (1993), 209-228.
3
[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,
10
104(2) (1999), 305-314.
11
[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
15
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).
18
[10] M. Junjun, Y. Dengbao and W. Cuicui, A novel cross-entropy and entropy measures of IFSs
19
and their applications, Knowledge-Based Systems, 48 (2013), 37-45.
20
[11] S. Kullback, Information theory and statistics, Dover publications, New York, USA, 1968.
21
[12] J. Lin, Divergence measures based on the Shannon entropy, IEEE Transactions on Informa-
22
tion Theory, 37(1) (1991), 145-151.
23
[13] S. Montes, I. Couso, P. Gil and C. Bertoluzza, Divergence measure between fuzzy sets, Inter-
24
national Journal of Approximate Reasoning, 30(2) (2002), 91-105.
25
[14] I. Montes, V. Janis and S. Montes, An axiomatic denition of divergence for intuitionistic
26
fuzzy sets, Advances in Intelligent Systems Research, EUSFLAT 2011, Atlantis Press, Aix-Les
27
Bains, ISBN 978-90-78677-00-0, (2011), 547-553.
28
[15] I. Montes, N. R. Pal, V. Janis and S. Montes, Divergence measures for intuitionistic fuzzy
29
sets, IEEE Transactions on Fuzzy Systems, 23 (2015) 444-456.
30
[16] G. A. Papakostas, A. G. Hatzimichailidis and V. G. Kaburlasos, Distance and similarity
31
measures between intuitionistic fuzzy sets: a comparative analysis from a pattern recognition
32
point of view, Pattern Recognition Letters, 34(14) (2013), 1609-1622.
33
[17] X. G. Shang, W. S. Jiang, A note on fuzzy information measures, Pattern Recognition
34
Letters, 18(5) (1997), 425-432
35
[18] E. Szmidt and J. Kacprzyk, Intuitionistic fuzzy sets in intelligent data analysis for medical
36
diagnosis, Proceedings of the Computational Science ICCS. Springer, Berlin, Germany, 2074,
37
(2001), 263-271.
38
[19] E. Szmidt and J. Kacprzyk, Intuitionistic fuzzy sets in some medical applications, Pro-
39
ceedings of the 7th Fuzzy Days, 2206 Computational intelligence: theory and applications.
40
Springer, Berlin, Germany (2001), 148-151.
41
[20] E. Szmidt and J. Kacprzyk, A Similarity Measure for Intuitionistic Fuzzy Sets and its Ap-
42
plication in Supporting Medical Diagnostic Reasoning, Articial Intelligence and Soft Com-
43
puting { ICAISC, 3070 (2004), 388-393.
44
[21] K. Vlachos and G. D. Sergiadis, Intuitionistic fuzzy information|applications to pattern
45
recognition, Pattern Recognition Letters, 28(2) (2007), 197-206.
46
[22] P. Wei and J. Ye, Improved intuitionistic fuzzy cross-entropy and its application to pattern
47
recognition, International Conference on Intelligent Systems and Knowledge Engineering,
48
(2010), 114-116.
49
[23] M. Xia and Z. Xu,Entropy/cross entropy-based group decision making under intuitionistic
50
fuzzy environment, Information Fusion, 13(1) (2012), 31-47.
51
[24] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338-353.
52
[25] Q. S. Zhang and S. Y. Jiang, A note on information entropy measures for vague sets and its
53
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
[1] S. Alonso, FJ. Cabrerizo, F. Chiclana, F. Herrera, E. Herrera-Viedma, Group decision mak-
1
ing with incomplete fuzzy linguistic preference relations, International Journal of Intelligent
2
Systems., 24(2) (2009), 201-222.
3
[2] M. Amiri-Aref, N. Javadian N, A new fuzzy TOPSIS method for material handling system
4
selection problems, Proceedings of the 8th WSEAS International Conference on Software
5
Engineering, Parallel and Distributed Systems., (2009), 169-174.
6
[3] KT. Atanassov Intuitionistic fuzzy sets, In: V. Sgurev (Ed.), VII ITKRs Session, Soa, June
7
1983 Central Sci. and Techn. Library, Bulg. Academy of Sciences., 1984.
8
[4] KT. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems., 20 (1986), 87-96.
9
[5] KT. Atanassov, New operations dened over the intuitionistic fuzzy sets, Fuzzy Sets and
10
Systems., 61 (1994), 137-142.
11
[6] KT. Atanassov, Intuitionistic Fuzzy Sets, Springer-Verlag, Heidelberg., 1999.
12
[7] KT. Atanassov, On Intuitionistic Fuzzy Sets Theory, Studies in Fuzziness and Soft Computing.
13
Springer-Verlag., 2012.
14
[8] KT. Atanassov and C. Georgiev, Intuitionistic fuzzy prolog, Fuzzy Sets and Systems., 53
15
(1993), 121-128.
16
[9] KT. Atanassov, NG. Nikolov and HT. Aladjov, Remark on two operations over intuitionistic
17
fuzzy sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems.,
18
9(1) (2003), 71-75.
19
[10] KT. Atanassov, G. Pasi and RR. Yager, Intuitionistic fuzzy interpretations of multi-criteria
20
multi-person and multi-measurement tool decision making, International Journal of Systems
21
Science., 36 (2005), 859-868.
22
[11] FE. Boran, S. Genc, M. Kurt and D. Akay, A multi-criteria intuitionistic fuzzy group decision
23
making for supplier selection with TOPSIS method, Expert Systems with Applications., 36
24
(2009), 11363-11368.
25
[12] P. Burillo and H. Bustince H, Entropy on intuitionistic fuzzy sets and on interval-valued
26
fuzzy sets, Fuzzy Sets and Systems., 78 (1996), 305-316.
27
[13] G. Buyukozkan and D. Ruan, Evaluation of software development projects using a fuzzy
28
multi-criteria decision approach, Mathematics and Computers in Simulation., 77 (2008),
29
[14] C-L. Chang and C-H. Hsu, Multi-criteria analysis via the VIKOR method for prioritizing
30
land-use restraint strategies in the Tseng-Wen reservoir watershed, Journal of Environmental
31
Management., 90 (2009), 3226-3230.
32
[15] LY. Chen and T-C. Wang, Optimizing partners choice in IS/IT outsourcing projects: The
33
strategic decision of fuzzy VIKOR, International Journal of Production Economics., 120
34
(2009), 233-242.
35
[16] SJ. Chen and CL. Hwang, Fuzzy multiple attribute decision making: methods and applica-
36
tions, Springer-Verlag, Berlin., 1992.
37
[17] T. Chen, Remarks on the Subtraction and Division Operations over Intuitionistic Fuzzy Sets
38
and Interval-Valued Fuzzy Sets, International Journal of Fuzzy Systems., 9 (2007), 169-172.
39
[18] R. De SK Biswas and AR. Roy, Some operations on intuitionistic fuzzy sets, Fuzzy Sets and
40
Systems., 114 (2000), 477-484.
41
[19] R. Degani and G. Bortolan, The problem of linguistic approximation in clinical decision
42
making, International Journal of Approximate Reasoning., 2 (1988), 143-162.
43
[20] M. Delgado, J. Verdegay and M. Vila, On aggregation operations of linguistic labels, International
44
Journal of Intelligent Systems., 8 (1993), 351-370.
45
[21] G. Deschrijver and EE. Kerre, On the representation of intuitionistic fuzzy t-norms and
46
t-conorms, IEEE Transactions on Fuzzy Systems., 12 (2004), 45-61.
47
[22] Y. Dong, Y. Xu and S. Yu, Computing the numerical scale of the linguistic term set for
48
the 2-tuple fuzzy linguistic representation model, IEEE Transactions on Fuzzy Systems., 17
49
(2009), 1366-1378.
50
[23] E. Gurkan, I. Erkmen and AM. Erkmen, Two-way fuzzy adaptive identication and control
51
exible-joint robot arm, Information Science., 145 (2002), 13-43.
52
[24] F. Herrera and E. Herrera-Viedma, Linguistic decision analysis: steps for solving decision
53
problems under linguistic information, Fuzzy Sets and Systems., 115 (2000), 67-82.
54
[25] F. Herrera, E. Herrera-Viedma and L. Martinez, A fuzzy linguistic methodology to deal with
55
unbalanced linguistic term sets, IEEE Transactions on Fuzzy Systems., 16 (2008), 354-370.
56
[26] F. Herrera and L. Martnez, A 2-tuple fuzzy linguistic representation model for computing
57
with words, IEEE Transactions on Fuzzy Systems., 8 (2000), 746- 752.
58
[27] WL. Hung and MS. Yang, Similarity measures of intuitionistic fuzzy sets based on Lp metric,
59
International Journal of Approximate Reasoning., 46 (2007), 120-136.
60
[28] M-S. Kuo, G-H. Tzeng and W-C. Huang Group decision-making based on concepts of ideal
61
and anti-ideal points in a fuzzy environment, Mathematical and Computer Modelling., 45
62
(2007), 324-339.
63
[29] D-F. Li, Compromise ratio method for fuzzy multi-attribute group decision making, Applied
64
Soft Computing., 7 (2006), 807-817.
65
[30] D-F. Li, Multiattribute group decision making method using extended linguistic variables
66
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems., 17 (2007),
67
[31] L. Lin, XH. Yuan and ZQ. Xia, M ulticriteria fuzzy decision-making methods based on intuitionistic
68
fuzzy sets, Journal of Computer and System Sciences., 73 (2007), 84-88.
69
[32] H. Malekly, SM. Mousavi and H. Hashemi, A fuzzy integrated methodology for evaluating
70
conceptual bridge design, Expert Systems with Applications., 37 (2010), 4910-4920.
71
[33] J. Mendel, LA. Zadeh, E. Trillas, R. Yager, J. Lawry, H. Hagras and S. Guadarrama What
72
computing with words means to me, IEEE Computational Intelligence Magazine., 5 (2010),
73
[34] L. Mikhailov and P. Tsvetinov, Evaluation of services using a fuzzy analytic hierarchy process,
74
Applied Soft Computing., 5 (2004), 23-33.
75
[35] SM. Mousavi SM, F. Jolai and R. Tavakkoli-Moghaddam, A fuzzy stochastic multi-attribute
76
group decision-making approach for selection problems, Group Decision and Negotiation., 22
77
(2004), 207-233.
78
[36] SM. Mousavi, SA. Torabi and R. Tavakkoli-Moghaddam, A hierarchical group decision-
79
making approach for new product selection in a fuzzy environment, Arabian Journal for
80
Science and Engineering., 38 (2013), 3233-3248.
81
[37] SM. Mousavi, B. Vahdani, R. Tavakkoli-Moghaddam, S. Ebrahimnejad and M. Amiri, A
82
multi-stage decision making process for multiple attributes analysis under an interval-valued
83
fuzzy environment, International Journal of Advanced Manufacturing Technology., 64 (2013),
84
1263-1273.
85
[38] S. Opricovic S, Multi-criteria optimization of civil engineering systems, Faculty of Civil
86
Engineering, Belgrade (1998).
87
[39] S. Opricovic S and G-H. Tzeng, Compromise solution by MCDM methods: A comparative
88
analysis of VIKOR and TOPSIS, European Journal of Operational Research., 156 (2004),
89
[40] S. Opricovic and G-H. Tzeng, Extended VIKOR method in comparison with outranking meth-
90
ods, European Journal of Operational Research., 178 (2007), 514-529.
91
[41] A. Sanayei, SF. Mousavi and A. Yazdankhah, Group decision making process for supplier se-
92
lection with VIKOR under fuzzy environment, Expert Systems with Applications., 37 (2010),
93
[42] MS. Shu, CH. Cheng and JR. Chang, Using intuitionistic fuzzy sets for faulttree analysis on
94
printed circuit board assembly, Microelectronics Reliability., 46 (2006), 2139-2148.
95
[43] E. Szmidt and J. Kacprzyk, Using intuitionistic fuzzy sets in group decision making, Control
96
Cybernetics., 31 (2002), 1037-1053.
97
[44] L-T. Tong, C-C. Chen and C-H. Wang, Optimization of multi-response processes using the
98
VIKOR method, International Journal of Advanced Manufacturing Technology., 31 (2007),
99
1049-1057.
100
[45] G-H. Tzeng, M-H. Teng, J-J. Chen and S. Opricovic, Multicriteria selection for a restaurant
101
location in Taipei, International Journal of Hospitality Management., 21 (2002), 171-187.
102
[46] B. Vahdani, SM. Mousavi, R. Tavakkoli-Moghaddam and R. Hashemi, A new design of the
103
elimination and choice translating reality method for multiple criteria group decision-making
104
in an intuitionistic fuzzy environment, Applied Mathematical Modelling., 37 (2013), 1781-
105
[47] B. Vahdani, R. Tavakkoli-Moghaddam, SM. Mousavi and A. Ghodratnama, Soft computing
106
based on new fuzzy modied multi-criteria decision making method, Applied Soft Computing.,
107
13 (2013), 165-172.
108
[48] B. Vahdani and H. Hadipour, Extension of the ELECTRE method based on interval-valued
109
fuzzy sets, Soft Computing., 15 (2011), 569-579.
110
[49] J. Wang and J. Hao J, A new version of 2-tuple fuzzy linguistic representation model for
111
computing with words, IEEE Transactions on Fuzzy Systems., 14 (2010), 435-445.
112
[50] X. Wang, Z. Gao and G. Wei, An approach to archives websites performance evaluation in
113
our country with interval intuitionistic fuzzy information, Advances in information sciences
114
and service sciences., 3 (2011), 112-117.
115
[51] Z. Xu Z, A method based on linguistic aggregation operators for group decision making with
116
linguistic preference relations, Information Sciences., 166 (2004), 19-30.
117
[52] ZS. Xu ZS, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems.,
118
15 (2007), 1179-1187.
119
[53] ZS. Xu and RR. Yager, Some geometric aggregation operators based on intuitionistic fuzzy
120
sets, International Journal of General Systems., 35 (2006), 417-433.
121
[54] R. Yager, A new methodology for ordinal multiobjective decisions based on fuzzy sets, Decision
122
Sciences., 12 (1981), 589-600.
123
[55] L. Zadeh, Fuzzy logic = computing with words, IEEE Transactions on Fuzzy Systems., bf 94
124
(1996), 103-111.
125
[56] L. Zadeh and J. Kacprzyk, Computing with words in information / Intelligent systems 1
126
(Foundations), Studies in Fuzziness and Soft Computing., 33 (1999), Springer-Verlag.
127
[57] L. Zadeh and J. Kacprzyk, Computing with words in information / Intelligent systems 2
128
(Applications), Studies in Fuzziness and Soft Computing., 34 (1999), Springer-Verlag.
129
[58] LA. Zadeh, Fuzzy sets, Information and Control., 8 (1965), 338-353.
130
[59] CY. Zhang and HY. Fu, Similarity measures on three kinds of fuzzy sets, Pattern Recognition
131
Letters., 27 (2006), 1307-1317.
132
[60] S-F. Zhang and S-Y Liu, A GRA-based intuitionistic fuzzy multi-criteria group decision
133
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
2017-09-23T11:23:20
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
2017-09-23T11:23:20
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
2017-09-23T11:23:20
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
2017-09-23T11:23:20
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
2017-09-23T11:23:20
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.
23
[12] Z. M. Ma and B. Q. Hu, Topological and lattice structures of L-fuzzy rough sets determined
24
by lower and upper sets, Information Sciences, 218 (2013), 194{204.
25
[13] N. N. Morsi and M. M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets Systems, 100
26
(1998), 327-342.
27
[14] Z. Pawlak, Rough sets, Inter. J. Comp. Info. Sci., 161 (2010), 2923-2944.
28
[15] K. Qin and Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems,
29
151 (2005), 601{613.
30
[16] K. Qin, J. Yang and Z. Pei, Generalized rough sets based on re
31
exive and transitive, Info.
32
Sci., 178 (2008), 4138{4141.
33
[17] A. M. Radzikowoska and E. E. Kerre, Fuzzy rough sets based on residuated lattices, In:
34
Transaction on Rough sets II, in Lincs, 3135 (2004), 278{296.
35
[18] A. A. Ramadan, Smooth topological Spaces, Fuzzy Sets and Systems, 48(3) (1992), 371{357.
36
[19] A. A. Ramadan, L-fuzzy interior systems, Comp. and Math. with Appl., 62 (2011), 4301{
37
[20] S. E. Rodabaugh and E. P. Kelment, Topological and algebraic structures in fuzzy sets, The
38
Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in Logic 20,
39
Kluwer Academic Publisher, Boston (2003).
40
[21] A. Sostak, On a fuzzy topological structure, Rend. Circ. Mat. Palermo (Supp. Ser.II), 11
41
(1985), 89{103.
42
[22] E. Turunen, Mathematics behind fuzzy logic, A Springer Verlag Co., Hiedelberg (1999).
43
[23] C. Y. Wang and B. Q. Hu, Fuzzy rough sets based on generalized residuated lattices, Information
44
Sciences, 248 (2013), 31{49.
45
[24] W. Z. Wu, A study on relationship between fuzzy rough approximation operators and fuzzy
46
topological spaces,
47
c Springer-Verlag, Berlin, Heidelberg (2005).
48
[25] W. Z. Wu, J. S. Mi and W. X. Zhang, Generalized fuzzy rough sets, Information Sciences,
49
151 (2003), 263{282.
50
[26] Y. Y. Yao, Constructive and Algebraic methods of the theory of rough sets, Information
51
Sciences, 109 (1998), 21{27.
52
[27] W. X. Zhang, Y. Leung and W. Z.Wu, Information systems and knowledge discovery, Science
53
Press,Beijing (2003).
54
[28] P. Zhi, P. Daowu and Z. Li, Topology vs generalized rough sets, Fuzzy Sets and Systems,
55
52(2) (2011), 231{239.
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.
27
[15] A. Dvurecenskij and O. Zahiri, Pseudo equality algebras: revision, Soft Computing, doi:
28
10.1007/s00500-015-1888-x, (2015).
29
[16] G. Georgescu and A. Iorgulescu, Pseudo MV-algebras, Multiple-Valued Logic, 6(1-2) (2001),
30
[17] G. Georgescu and A. Iorgulescu, Pseudo-BCK algebras: An extension of BCK-algebras, Proceedings
31
of DMTCS'01: Combinatorics, Computability and Logic, Springer, London, (2001),
32
[18] Y. Imai and K. Iseki, On axiom systems of propositional calculi XIV, Proceedings of the
33
Japan Academy, 42(1) (1966), 19-22.
34
[19] A. Iorgulescu, Classes of pseudo-BCK algebras - Part I, Journal of Multiple-Valued Logic
35
and Soft Computing, 12(1-2) (2006), 71-130.
36
[20] A. Iorgulescu, Algebras of logic as BCK-algebras, ASE Ed., Bucharest, 2008.
37
[21] H. S. Kim and Y. H. Kim, On BE-algebras, Scientiae Mathematicae Japonicae, 66(1) (2007),
38
[22] K. H. Kim and Y. H. Yon, Dual BCK-algebra and MV-algebra, Scientiae Mathematicae
39
Japonicae, 66(2) (2007), 247-254.
40
[23] J. Kuhr, Pseudo-BCK semilattices, Demonstratio Mathematica, 40(3) (2007), 495-516.
41
[24] J. Kuhr, Pseudo-BCK algebras and related structures, Habilitation thesis, Palacky University
42
in Olomouc, 2007.
43
[25] J. Kuhr, Commutative pseudo BCK-algebras, Southeast Asian Bulletin of Mathematics,
44
33(3) (2009), 451-475.
45
[26] K. J. Lee, Pseudo-valuations on BE-algebras, Applied Mathematical Sciences, 7(125) (2013),
46
6199-6207.
47
[27] B. L. Meng, CI-algebras, Scientiae Mathematicae Japonicae, 71(1) (2010), 11-17.
48
[28] B. L. Meng, On lters in BE-algebras, Scientiae Mathematicae Japonicae, 71(2) (2010),
49
[29] J. Rachunek, A non-commutative generalization of MV-algebras, Czechoslovak Mathematical
50
Journal, 52(127) (2002), 255-273.
51
[30] A. Rezaei and A. Borumand Saeid, On fuzzy subalgebras of BE-algebras, Afrika Matematika,
52
22(2) (2011), 115-127.
53
[31] A. Rezaei and A. Borumand Saeid, Some results in BE-algebras, Annals of Oradea University
54
- Mathematics Fascicola, XIX(1) (2012), 33-44.
55
[32] A. Rezaei and A. Borumand Saeid, Commutative ideals in BE-algebras, Kyungpook Mathematical
56
Journal, 52(4) (2012), 483-494.
57
[33] A. Rezaei, A. Borumand Saeid and R. A. Borzooei, Relation between Hilbert algebras and
58
BE-algebras, Applications and Applied Mathematics, 8(2) (2013), 573-584.
59
[34] A. Rezaei, A. Borumand Saeid, A. Radfar and R. A. Borzooei, Congruence relations on
60
pseudo BE-algebras, Annnals of the University of Craiova, Mathematics and Computer Sciences
61
Series, 41(2) (2014), 166-176.
62
[35] A. Rezaei, L. C. Ciungu and A. Borumand Saeid, States on pseudo BE-algebras, submitted.
63
[36] A. Walendziak, On commutative BE-algebras, Scientiae Mathematicae Japonicae, 69(2)
64
(2009), 281-284.
65
[37] A. Walendziak, On normal lters and congruence relations in BE-algebras, Commentationes
66
Mathematicae, 52(2) (2012), 199-205.
67
[38] H. Zhou and B. Zhao, Generalized Bosbach and Riecan states based on relative negations in
68
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
AUTHOR
[1] R. Balbes and Ph. Dwinger, Distributive lattices, University of Missouri Press, 1974.
1
[2] W. J. Blok and D. Pigozzi, Algebraizable logics, Memoirs of the American Mathematical
2
Society, Amer. Math. Soc, Providence, 396 (1989).
3
[3] K. Blount and C. Tsinakis, The structure of residuated lattices, Internat. J. Algebra Comput.,
4
13(4) (2003), 437-461.
5
[4] D. Busneag and D. Piciu, Some types of lters in residuated lattices, Soft Comput., 18(5)
6
(2014), 825-837.
7
[5] D. Busneag and D. Piciu, A new approach for classication of lters in residuated lattices,
8
Fuzzy Sets and Systems, 260 (2015), 121-130.
9
[6] D. Busneag, D. Piciu and L. Holdon, Some properties of ideals in Stonean Residuated Lattices,
10
J. Multi-Valued Logic & Soft Computing, 24(5-6) (2015), 529-546.
11
[7] D. Busneag, D. Piciu and J. Paralescu, Divisible and semi-divisible residuated lattices, Ann.
12
St. Univ. Al. I. Cuza, Iasi, Matematica (S.N.), doi:10.2478/aicu-2013-0012, (2013), 14-45.
13
[8] C. C. Chang, Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958),
14
[9] L. Chun-hui and X. Luo-shan, On -Ideals and lattices of -Ideals in regular residuated
15
lattices, In B.-Y. Cao et al. (Eds.): Quantitative Logic and Soft Computing (2010), AISC
16
82, 425-434.
17
[10] R. Cignoli, I. M. L. D'Ottaviano and D. Mundici, Algebraic foundations of many-valued
18
reasoning, Trends in Logic-Studia Logica Library 7, Dordrecht: Kluwer Academic Publishers
19
[11] R. P. Dilworth, Non-commutative residuated lattices, Trans. Amer. Math. Soc., 46 (1939),
20
[12] P. Hajek, Metamathematics of fuzzy logic, Trends in Logic-Studia Logica Library 4, Dordrecht:
21
Kluwer Academic Publishers (1998).
22
[13] M. Haveshki, A. Borumand Saeid and E. Eslami, Some types of lters in BL-algebras, Soft
23
Comput., 10 (2010), 657-664.
24
[14] U. Hohle, Commutative residuated monoids, In: U. Hohle, P. Klement (eds), Non-classical
25
Logics and Their Aplications to Fuzzy Subsets, Kluwer Academic Publishers, (1995).
26
[15] P. M. Idziak, Lattice operations in BCK-algebras, Math. Japonica, 29 (1984), 839-846.
27
[16] A. Iorgulescu, Algebras of logic as BCK algebras, Ed. ASE, Bucuresti, 2008.
28
[17] M. Kondo and W. A. Dudek, Filter theory of BL-algebras, Soft Comput., 12 (2008), 419-423.
29
[18] W. Krull, Axiomatische Begrundung der allgemeinen Ideal theorie, Sitzungsberichte der
30
physikalisch medizinischen Societad der Erlangen, 56 (1924), 47-63.
31
[19] L. Lianzhen and L. Kaitai, Boolean lters and positive implicative lters of residuated lattices,
32
Inf. Sciences, 177 (2007), 5725-5738.
33
[20] X. Ma, J. Zhan and W. A. Dudek, Some kinds of (; _ q)-fuzzy lters of BL-algebras,
34
Computers and Mathematics with Applications, 58 (2009), 248-256.
35
[21] M. Okada and K. Terui, The nite model property for various fragments of intuitionistic
36
linear logic, Journal of Symbolic Logic, 64 (1999), 790-802.
37
[22] J. Pavelka, On fuzzy logic II. Enriched residuated lattices and semantics of propositional
38
calculi, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 25 (1979),
39
[23] D. Piciu, Algebras of fuzzy logic, Ed. Universitaria, Craiova (2007).
40
[24] E. Turunen, Boolean deductive systems of BL algebras, Arch. Math. Logic, 40 (2001).
41
[25] E. Turunen, Mathematics behind fuzzy logic, Physica-Verlag (1999).
42
[26] B. Van Gasse, G. Deschrijver, C. Cornelis and E. E. Kerre, Filters of residuated lattices and
43
triangle algebras, Inf. Sciences, 180(16) (2010), 3006-3020.
44
[27] M. Ward, Residuated distributive lattices, Duke Mathematcal Journal, 6 (1940), 641-651.
45
[28] M. Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc., 45 (1939), 335-
46
[29] M. A. Zhenming, MTL -lters and their characterization in residuated lattices, Computer
47
Engineering and Applications, 48(20) (2012), 64-66.
48
[30] Y. Zhu and Y. Xu, On lter theory of residuated lattices, Inf. Sciences, 180 (2010), 3614-3632.
49
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