Generalized ordinal sums of aggregation operators on bounded lattices. (English) Zbl 1459.68207
Summary: In this paper, four kinds of generalized ordinal sums of aggregation operators on bounded lattices are provided and discussed. Firstly, necessary and sufficient conditions for lower (resp. upper) generalized ordinal sum of many summands to be an aggregation operator are provided, in which case that the smallest (resp. greatest) aggregation operator that coincides with each summand is given. Meanwhile, the idempotent lower (resp. upper) generalized ordinal sum of idempotent aggregation operators is also discussed. Then the relationships between these four kinds of generalized ordinal sums are given. Finally, the lattice structures of some sets of special aggregation operators are investigated.
MSC:
| 68T37 | Reasoning under uncertainty in the context of artificial intelligence |
Keywords:
aggregation operator; lower generalized ordinal sum; idempotent lower generalized ordinal sum; bounded latticeReferences:
| [1] | Aczél, J., Lectures on Functional Equations and Their Applications (1966), Academic Press: Academic Press New York · Zbl 0139.09301 |
| [2] | Beliakov, G.; Pradera, A.; Calvo, T., Aggregation Functions: A Guide for Practitioners (2007), Springer-Verlag: Springer-Verlag Berlin · Zbl 1123.68124 |
| [3] | Birkhoff, G., Lattice Theory (1967), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 0126.03801 |
| [4] | Blok, W. J.; Ferreirim, I. M.A., On the structure of hoops, Algebra Universalis, 43, 233-257 (2000) · Zbl 1012.06016 |
| [5] | Bustince, H.; Fernandez, J.; Mesiar, R.; Montero, J.; Orduna, R., Overlap functions, Nonlinear Anal. Theory Methods Appl., 72, 1488-1499 (2010) · Zbl 1182.26076 |
| [6] | Bustince, H.; Pagola, M.; Mesiar, R.; Hüllermeier, E.; Herrera, F., Grouping, overlap, and generalized bientropic functions for fuzzy modeling of pairwise comparisons, IEEE Trans. Fuzzy Syst., 20, 405-415 (2012) |
| [7] | Calvo, T.; De Baets, B.; Fodor, J., The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets Syst., 120, 385-394 (2001) · Zbl 0977.03026 |
| [8] | Çaylı, G. D.; Karaçal, F.; Mesiar, R., On a new class of uninorms on bounded lattices, Inf. Sci., 367-368, 221-231 (2016) · Zbl 1521.03211 |
| [9] | Clifford, A. H., Naturally totally ordered commutative semigroups, Am. J. Math., 76, 631-646 (1954) · Zbl 0055.01503 |
| [10] | Dan, Y. X.; Hu, B. Q.; Qiao, J. S., New construction of t-norms and t-conorms on bounded lattices, Fuzzy Sets Syst. (2019) |
| [11] | Davey, B. A.; Priestley, H. A., Introduction to Lattices and Order (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1002.06001 |
| [12] | De Baets, B., Idempotent uninorms, Eur. J. Oper. Res., 118, 631-642 (1999) · Zbl 0933.03071 |
| [13] | De Baets, B.; Mesiar, R., Ordinal sums of aggregation operators, (Bouchon-Meunier, B.; Gutiérrez-Ríos, J.; Magdalena, L.; Yager, R. R., Technologies for Constructing Intelligent Systems 2: Tools (2002), Springer-Verlag: Springer-Verlag Berlin), 137-147 · Zbl 1015.68194 |
| [14] | Dimuro, G. P.; Bedregal, B., Archimedean overlap functions: the ordinal sum and the cancellation, idempotency and limiting properties, Fuzzy Sets Syst., 252, 39-54 (2014) · Zbl 1334.68217 |
| [15] | Dubois, D.; Prade, H., On the use of aggregation operations in information fusion processes, Fuzzy Sets Syst., 142, 143-161 (2004) · Zbl 1091.68107 |
| [16] | El-Zekey, M., Lattice-based sum of t-norms on bounded lattices, Fuzzy Sets Syst., 386, 60-76 (2020) · Zbl 1465.03084 |
| [17] | El-Zekey, M.; Medina, J.; Mesiar, R., Lattice-based sums, Inf. Sci., 223, 270-284 (2013) · Zbl 1293.06002 |
| [18] | Ertuğrul, Ü.; Karaçal, F.; Mesiar, R., Modified ordinal sums of triangular norms and triangular conorms on bounded lattices, Int. J. Intell. Syst., 30, 807-817 (2015) |
| [19] | Fodor, J.; Roubens, M., Fuzzy Preference Modelling and Multicriteria Decision Support (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0827.90002 |
| [20] | Fodor, J.; Yager, R. R.; Rybalov, A., Structure of uninorms, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 5, 411-427 (1997) · Zbl 1232.03015 |
| [21] | Goguen, J. A., L-fuzzy sets, J. Math. Anal. Appl., 18, 145-174 (1967) · Zbl 0145.24404 |
| [22] | Grabisch, M.; Marichal, J.-L.; Mesiar, R.; Pap, E., Aggregation Functions (2009), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1196.00002 |
| [23] | Jenei, S., A note on the ordinal sum theorem and its consequence for the construction of triangular norms, Fuzzy Sets Syst., 126, 199-205 (2002) · Zbl 0996.03508 |
| [24] | Karaçal, F.; Ertuğrul, Ü.; Mesiar, R., Characterization of uninorms on bounded lattices, Fuzzy Sets Syst., 308, 54-71 (2017) · Zbl 1393.03032 |
| [25] | Karaçal, F.; Mesiar, R., Aggregation functions on bounded lattices, Int. J. Gen. Syst., 46, 37-51 (2017) |
| [26] | Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0972.03002 |
| [27] | Komorníková, M.; Mesiar, R., Aggregation functions on bounded partially ordered sets and their classification, Fuzzy Sets Syst., 175, 48-56 (2011) · Zbl 1253.06004 |
| [28] | Ling, C.-H., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212 (1995) · Zbl 0137.26401 |
| [29] | Liu, H. W., Distributivity and conditional distributivity of semi-uninorms over continuous t-conorms and t-norms, Fuzzy Sets Syst., 268, 27-43 (2015) · Zbl 1361.03021 |
| [30] | Medina, J., Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices, Fuzzy Sets Syst., 202, 75-88 (2012) · Zbl 1272.03112 |
| [31] | Mesiar, R.; Sempi, C., Ordinal sums and idempotents of copulas, Aequationes Math., 79, 39-52 (2010) · Zbl 1205.62063 |
| [32] | Nelsen, R. B., An Introduction to Copulas, Lecture Notes in Statistics, 139 (1999), Springer: Springer New York · Zbl 0909.62052 |
| [33] | Qin, F.; Zhao, B., The distributive equations for idempotent uninorms and nullnorms, Fuzzy Sets Syst., 155, 446-458 (2005) · Zbl 1077.03514 |
| [34] | Rudas, I. J.; Pap, E.; Fodor, J., Information aggregation in intelligent systems: an application oriented approach, Knowl.-Based Syst., 38, 3-13 (2013) |
| [35] | Saminger, S., On ordinal sums of triangular norms on bounded lattices, Fuzzy Sets Syst., 157, 1403-1416 (2006) · Zbl 1099.06004 |
| [36] | Yager, R. R.; Rybalov, A., Uninorm aggregation operators, Fuzzy Sets Syst., 80, 111-120 (1996) · Zbl 0871.04007 |
| [37] | Zhang, D. X., Triangular norms on partially ordered sets, Fuzzy Sets Syst., 153, 195-209 (2005) · Zbl 1091.03025 |
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.
