Bedregal, Benjamín; Beliakov, Gleb; Bustince, Humberto; Calvo, Tomasa; Mesiar, Radko; Paternain, Daniel A class of fuzzy multisets with a fixed number of memberships. (English) Zbl 1247.03113 Inf. Sci. 189, 1-17 (2012). Summary: The main aim of this work is to present a generalization of Atanassov’s operators to higher dimensions. To do so, we use the concept of fuzzy set, which can be seen as a special kind of fuzzy multiset, to define a generalization of Atanassov’s operators for \(n\)-dimensional fuzzy values (called \(n\)-dimensional intervals). We prove that our generalized Atanassov’s operators also generalize OWA operators of any dimension by allowing negative weights. We apply our results to a decision making problem. We also extend the notions of aggregating functions, in particular t-norms, fuzzy negations and automorphism and related notions for \(n\)-dimensional framework. Cited in 24 Documents MSC: 03E72 Theory of fuzzy sets, etc. Keywords:fuzzy multisets; \(n\)-dimensional fuzzy sets; Atanassov operators; OWA operator; \(n\)-dimensional aggregation functions; \(n\)-dimensional negation PDFBibTeX XMLCite \textit{B. Bedregal} et al., Inf. Sci. 189, 1--17 (2012; Zbl 1247.03113) Full Text: DOI References: [1] Atanassov, K. T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96 (1986) · Zbl 0631.03040 [2] Atanassov, K. T.; Gargov, G., Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31, 343-349 (1989) · Zbl 0674.03017 [3] Atanassov, K., Intuitionistic Fuzzy Sets, Theory and Applications (1999), Physica-Verlag: Physica-Verlag Heidelberg · Zbl 0939.03057 [4] Bedregal, B. C., On interval fuzzy negations, Fuzzy Sets and Systems, 161, 2290-2313 (2010) · Zbl 1204.03030 [5] Bedregal, B. C.; Dimuro, G. P.; Santiago, R. H.N.; Reiser, R. H.S., On interval fuzzy \(S\)-implications, Information Sciences, 180, 1373-1389 (2010) · Zbl 1189.03027 [6] Bedregal, B. C.; Takahashi, A., The best interval representation of t-norms and automorphisms, Fuzzy Sets and Systems, 157, 24, 3220-3230 (2006) · Zbl 1114.03040 [7] Beliakov, G.; Pradera, A.; Calvo, T., Aggregation Functions: A Guide for Practitioners (2007), Springer: Springer Berlin · Zbl 1123.68124 [8] Bustince, H.; Barrenechea, E.; Pagola, M., Generation of interval-valued fuzzy and Atanassov’s intuitionistic fuzzy connectives from fuzzy connectives and from \(K_α\) operators: laws for conjunctions and disjunctions, amplitude, International Journal of Intelligent Systems, 23, 680-714 (2008) · Zbl 1140.68499 [9] Bustince, H.; Barrenechea, E.; Pagola, M.; Fernandez, J., Interval-valued fuzzy sets constructed from matrices: application to edge detection, Fuzzy Sets and Systems, 160, 1819-1840 (2009) · Zbl 1182.68191 [10] Bustince, H.; Calvo, T.; De Baets, B.; Fodor, J.; Mesiar, R.; Montero, J.; Paternain, D.; Pradera, A., A class of aggregation operators encompassing two-dimensional OWA operators, Information Sciences, 180, 1977-1989 (2010) · Zbl 1205.68419 [11] Bustince, H.; Montero, J.; Pagola, M.; Barrenechea, E.; Gomez, D., A survey of interval-valued fuzzy sets, (Pedrycz, W.; Skowron, A.; Kreinovich, V., Handbook of Granular Computing (2008), John Wiley & Sons, Ltd.: John Wiley & Sons, Ltd. West Sussex), 491-515, Chapter 22 [12] Bustince, H.; Pagola, M.; Barrenechea, E.; Fernandez, J.; Melo-Pinto, P.; Couto, P.; Tizhoosh, H.; Montero, J., Ignorance functions. An application to the calculation of the threshold in prostate ultrasound images, Fuzzy Sets and Systems, 161, 20-36 (2010) [13] Da Costa, C. G.; Bedregal, B. C.; Doria Neto, A. D., Relating De Morgan triples with Atanassov’s intuitionistic De Morgan triples via automorphisms, International Journal of Approximate Reasoning, 52, 4, 473-487 (2011) · Zbl 1228.03038 [14] Deschrijver, G.; Cornelis, C., Representability in interval-valued fuzzy set theory, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 15, 3, 345-361 (2007) · Zbl 1144.03033 [15] Deschrijver, G., A representation of t-norms in interval-valued \(L\)-fuzzy set theory, Fuzzy Sets and Systems, 159, 1597-1618 (2008) · Zbl 1176.03027 [16] Deschrijver, G.; Kerre, E., On the relation between some extensions of fuzzy set theory, Fuzzy Sets and Systems, 133, 227-235 (2003) · Zbl 1013.03065 [17] Deschrijver, G.; Kerre, E., Aggregation operation in interval-valued fuzzy and Atanassov’s intuitionistic fuzzy set theory, (Bustince, H.; etal., Fuzzy Sets and Their Extensions: Representation, Aggregation and Models (2008), Springer: Springer Berlin) · Zbl 1370.03070 [18] Dubois, D.; Prade, H., Bipolar representations: 1. Cognition and decision, 2. Reasoning and learning, International Journal of Intelligent Systems, 23, 8-10 (2008) [19] Gehrke, M.; Walker, C.; Walker, E., Some comments on interval valued fuzzy sets, International Journal of Intelligent Systems, 11, 751-759 (1996) · Zbl 0865.04006 [20] Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., Continuous Lattices and Domains (2003), Cambridge University Press: Cambridge University Press Cambridge [21] Goguen, J., L-fuzzy sets, Journal of Mathematics Analysis Applied, 18, 145-167 (1967) · Zbl 0145.24404 [22] Grabisch, M.; Labreuche, C., Bi-capacities - I: definition, Moobius transform and interaction, Fuzzy Sets and Systems, 151, 2, 211-236 (2005) · Zbl 1106.91023 [23] Grabisch, M.; Marichal, J. L.; Mesiar, R.; Pap, E., Aggregation Functions (2009), Cambridge University Press: Cambridge University Press Cambridge [24] Grabisch, M., The lattice of embedded subsets, Discrete Applied Mathematics, 158, 5, 479-488 (2010) · Zbl 1186.91025 [25] Hüllermeier, E.; Brinker, K., Learning valued preference structures for solving classification problems, Fuzzy Sets and Systems, 159, 18, 2337-2352 (2008) · Zbl 1187.68394 [26] Hüllermeier, E.; Vanderlooy, S., Combining predictions in pairwise classification: an optimal adaptive voting strategy and its relation to weighted voting, Pattern Recognition, 43, 1, 128-142 (2010) · Zbl 1191.68578 [27] Hurwicz, L., Aggregation in macroeconomic models, Econometrica, 489-490 (1952) [28] Jenei, S., A more efficient method for defining fuzzy connectives, Fuzzy Sets and Systems, 90, 25-35 (1997) · Zbl 0922.03074 [29] Jurio, A.; Pagola, M.; Mesiar, R.; Beliakov, G.; Bustince, H., Image magnification using interval information, IEEE Transactions on Image Processing, 20, 11, 3112-3123 (2011) · Zbl 1372.94126 [30] Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms (2000), Kluwer: Kluwer Dordrecht · Zbl 0972.03002 [31] Mesiar, R.; Kolesárová, A.; Calvo, T.; Monorníková, M., A review of aggregation functions, (Bustince, H.; etal., Fuzzy Sets and Their Extensions: Representation, Aggregation and Models (2008), Springer: Springer Berlin), 121-144 · Zbl 1147.68081 [32] Mesiar, R.; Komorníková, M., Aggregation functions on bounded posets, (Cornelis, C.; etal., 35 Year of Fuzzy Set Theory: Celebration Volume Dedicated to the Retirement of Etienne E. Kerre (2010), Springer: Springer Berlin), 3-17 · Zbl 1231.03048 [33] Miyamoto, S., Multisets and fuzzy multisets, (Liu, Z.-Q.; Miyamoto, S., Soft Computing and Human-Centered Machines (2000), Springer: Springer Berlin), 9-33 · Zbl 0961.03048 [34] Moore, R. E.; Kearfott, R. B.; Cloud, M. J., Introduction to Interval Analysis (2009), SIAM: SIAM Philadelphia · Zbl 1168.65002 [35] Reiser, R. H.S.; Dimuro, G. P.; Bedregal, B. C.; Santiago, R. H.N., Interval valued QL-implications, (Leivant, D.; de Queiroz, R., WoLLIC 2007. WoLLIC 2007, Lectures Notes in Computer Science - LNCS, vol. 4576 (2007), Springer-Verlag: Springer-Verlag Berlin, Heildelberg), 307-321 · Zbl 1213.03036 [36] Sanz, J. A.; Fernandez, A.; Bustince, H.; Herrera, F., Improving the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets and genetic amplitude tuning, Information Sciences, 180, 3674-3685 (2010) [37] Sanz, J. A.; Fernandez, A.; Bustince, H.; Herrera, F., A genetic tuning to improve the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets: degree of ignorance and lateral position, International Journal of Approximate Reasoning, 52, 6, 751-766 (2011) [38] Scott, D. S., Continuous lattices, Lecture Notes in Mathematics, 274, 97-136 (1972) [39] Shang, Y.; Yuan, X.; Lee, E. S., The \(n\)-dimensional fuzzy sets and Zadeh fuzzy sets based on the finite valued fuzzy sets, Computers & Mathematics with Applications, 60, 442-463 (2010) · Zbl 1201.03048 [40] Trillas, E., Sobre funciones de negación en la teoria de los conjuntos difusos, Sthocastica, 3, 47-60 (1979) [41] Verma, A. K.; Verma, R.; Mahanti, N. C., Facility location selection: an interval-valued intuitionistic fuzzy TOPSIS approach, Modern Mathematics and Statistics, 4, 2, 68-72 (2010) [42] Wang, Z.; Li, K. W.; Wang, W., An approach to multiatribute decision making with interval-valued intuitionistic fuzzy assessments and incomplete weights, Information Sciences, 179, 3026-3040 (2009) · Zbl 1170.90427 [43] Wang, T. C.; Lin, Y. L., Applying the consistent fuzzy preference relations to select merger strategy for commercial banks in new financial environments, Expert Systems with Applications, 36, 7019-7026 (2009) [44] Xu, Z. S., On similarity measure of interval-valued intuitionistic fuzzy sets and their application to pattern recognitions, Journal of Southeast University, 23, 139-143 (2007) · Zbl 1141.03339 [45] Xu, Z. S., An approach to group decision making based on interval-valued intuitionistic judgment matrices, Systems Engineering - Theory and Practice, 27, 4, 126-133 (2007) [46] Xu, Z. S., A methods based on distance measure for interval-valued intuitionistic fuzzy group decision making, Information Sciences, 180, 181-190 (2010) · Zbl 1183.91039 [47] Yager, R. R., On the theory of bags, International Journal of General Systems, 13, 23-37 (1986) [48] Zadeh, L. A., Fuzzy sets, Information and Control, 338-353 (1965) · Zbl 0139.24606 [49] Zhang, W. R.; Zhang, L., Yin Yang bipolar logic and bipolar fuzzy logic, Information Sciences, 165, 265-287 (2004) · Zbl 1068.03020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.