×

Overlap and grouping functions on complete lattices. (English) Zbl 1497.06005

Summary: Recently, R. Paiva et al. [“Lattice-valued overlap and quasi-overlap functions”, Preprint, arXiv:1902.00133] introduced the concepts of lattice-valued overlap and quasi-overlap functions, and showed the migrativity, homogeneity and other properties of (quasi-) overlap functions on bounded lattices. In this paper, we continue to consider this research topic and study overlap and grouping functions on complete lattices in order to extend the continuity of these two operators from the unit closed interval to the lattices status by using join-preserving and meet-preserving properties of binary operators on complete lattices. More precisely, firstly, we introduce the notion of overlap functions on complete lattices and give two construction methods of them. Secondly, we show some basic properties of overlap functions on complete lattices. In particular, we introduce the concept of \((\wedge, \vee)\)-combination of overlap functions and extend the notions of migrativity and homogeneity of overlap functions on bounded lattices to the so-called \((\alpha, B, C)\)-migrativity and \((B, C)\)-homogeneity of overlap functions on complete lattices, respectively, where \(\alpha\) belongs to the complete lattice and \(B\) and \(C\) are two binary operators on the complete lattice, and then we focus on these properties along with the cancellation law of overlap functions on complete lattices. Finally, we give an analogous discussion for grouping functions on complete lattices.

MSC:

06B23 Complete lattices, completions
03E72 Theory of fuzzy sets, etc.
06D72 Fuzzy lattices (soft algebras) and related topics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bedregal, B.; Bustince, H.; Palmeira, E.; Dimuro, G.; Fernandez, J., Generalized interval-valued OWA operators with interval weights derived from interval-valued overlap functions, Int. J. Approximate Reasoning, 90, 1-16 (2017) · Zbl 1419.68154
[2] Bedregal, B.; Dimuro, G. P.; Bustince, H.; Barrenechea, E., New results on overlap and grouping functions, Inf. Sci., 249, 148-170 (2013) · Zbl 1335.68264
[3] Beliakov, G.; Pradera, A.; Calvo, T., Aggregation Functions: A Guide for Practitioners (2007), Springer: Springer Berlin · Zbl 1123.68124
[4] H. Bustince, J. Fernández, R. Mesiar, J. Montero, R. Orduna, Overlap index, overlap functions and migrativity, in: Proceedings of IFSA/EUSFLAT Conference, 2009, pp. 300-305. · Zbl 1182.26076
[5] Bustince, H.; Fernandez, J.; Mesiar, R.; Montero, J.; Orduna, R., Overlap functions, Nonlinear Anal., 72, 1488-1499 (2010) · Zbl 1182.26076
[6] Bustince, H.; Pagola, M.; Mesiar, R.; Hüllermeier, E.; Herrera, F., Grouping, overlaps, and generalized bientropic functions for fuzzy modeling of pairwise comparisons, IEEE Trans. Fuzzy Syst., 20, 405-415 (2012)
[7] Çaylí, G. D., On a new class of t-norms and t-conorms on bounded lattices, Fuzzy Sets Syst., 332, 129-143 (2018) · Zbl 1380.03042
[8] Dan, Y.; Hu, B. Q., A new structure for uninorms on bounded lattices, Fuzzy Sets Syst., 386, 77-94 (2020) · Zbl 1465.03083
[9] Dan, Y.; Hu, B. Q.; Qiao, J., General L-fuzzy aggregation functions based on complete residuated lattices, Soft. Comput., 24, 3087-3112 (2020) · Zbl 1436.03320
[10] De Baets, B.; Mesiar, R., Triangular norms on product lattices, Fuzzy Sets Syst., 104, 61-75 (1999) · Zbl 0935.03060
[11] De Miguel, L.; Gómez, D.; Rodríguez, J. T.; Montero, J.; Bustince, H.; Dimuro, G. P.; Sanz, J. A., General overlap functions, Fuzzy Sets Syst., 372, 81-96 (2019) · Zbl 1423.03208
[12] 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
[13] Dimuro, G. P.; Bedregal, B., On residual implications derived from overlap functions, Inf. Sci., 312, 78-88 (2015) · Zbl 1387.03022
[14] Dimuro, G. P.; Bedregal, B.; Bustince, H.; Jurio, A.; Baczyński, M.; Miś, K., QL-operations and QL-implication functions constructed from tuples (O, G, N))and the generation of fuzzy subsethood and entropy measures, Int. J. Approximate Reasoning, 82, 170-192 (2017) · Zbl 1452.03074
[15] Dimuro, G. P.; Bedregal, B.; Santiago, R. H.N., On (G, N))implications derived from grouping functions, Inf. Sci., 279, 1-17 (2014) · Zbl 1354.03030
[16] Dimuro, G. P.; Bedregal, B.; Bustince, H.; Asiáin, M. J.; Mesiar, R., On additive generators of overlap functions, Fuzzy Sets Syst., 287, 76-96 (2016) · Zbl 1392.68407
[17] Elkano, M.; Galar, M.; Sanz, J.; Fernández, A.; Barrenechea, E.; Herrera, F.; Bustince, H., Enhancing multi-class classification in FARC-HD fuzzy classifier: On the synergy between n-dimensional overlap functions and decomposition strategies, IEEE Trans. Fuzzy Syst., 23, 1562-1580 (2015)
[18] Elkano, M.; Galar, M.; Sanz, J. A.; Schiavo, P. F.; Pereira, S.; Dimuro, G. P.; Borges, E. N.; Bustince, H., Consensus via penalty functions for decision making in ensembles in fuzzy rule-based classification systems, Appl. Soft Comput., 67, 728-740 (2018)
[19] El-Zekey, M., Lattice-based sum of t-norms on bounded lattices, Fuzzy Sets Syst., 386, 60-76 (2020) · Zbl 1465.03084
[20] Ertuğrul, Ü., Construction of nullnorms on bounded lattices and an equivalence relation on nullnorms, Fuzzy Sets Syst., 334, 94-109 (2018) · Zbl 1380.03045
[21] 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)
[22] Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S., Continuous Lattices and Domains (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1088.06001
[23] Gómez, D.; Montero, J., A discussion on aggregation functions, Kybernetika, 40, 107-120 (2004) · Zbl 1249.68229
[24] Gómez, D.; Rodríguez, J. T.; Montero, J.; Bustince, H.; Barrenechea, E., n-dimensional overlap functions, Fuzzy Sets Syst., 287, 57-75 (2016) · Zbl 1392.68401
[25] Gómez, D.; Rodríguez, J. T.; Yáñez, J.; Montero, J., A new modularity measure for Fuzzy Community detection problems based on overlap and grouping functions, Int. J. Approximate Reasoning, 74, 88-107 (2016) · Zbl 1357.68239
[26] Gutiérrez García, J.; Höhle, U.; de Prada Vicente, M. A., On lattice-valued frames: the completely distributive case, Fuzzy Sets Syst., 161, 1022-1030 (2010) · Zbl 1191.06006
[27] Jurio, A.; Bustince, H.; Pagola, M.; Pradera, A.; Yager, R., Some properties of overlap and grouping functions and their application to image thresholding, Fuzzy Sets Syst., 229, 69-90 (2013) · Zbl 1284.68549
[28] Karaçal, F., On the direct decomposability of strong negations and S-implication operators on product lattices, Inf. Sci., 176, 3011-3025 (2006) · Zbl 1104.03016
[29] Karaçal, F.; İnce, M. A.; Mesiar, R., Nullnorms on bounded lattices, Inf. Sci., 325, 227-236 (2015) · Zbl 1387.03047
[30] Karaçal, F.; Mesiar, R., Uninorms on bounded lattices, Fuzzy Sets Syst., 261, 33-43 (2015) · Zbl 1366.03229
[31] Liu, H.; Zhao, B., On distributivity equations of implications over overlap functions and contrapositive symmetry equations of implications, J. Intell. Fuzzy Syst., 36, 283-294 (2019)
[32] Lopez-Molina, C.; De Baets, B.; Bustince, H.; Induráin, E.; Stupňanová, A.; Mesiar, R., Bimigrativity of binary aggregation functions, Inf. Sci., 274, 225-235 (2014) · Zbl 1341.68262
[33] Lucca, G.; Sanz, J. A.; Dimuro, G. P.; Bedregal, B.; Asiáin, M. J.; Elkano, M.; Bustince, H., CC-integrals: Choquet-like Copula-based aggregation functions and its application in fuzzy rule-based classification systems, Knowl.-Based Syst., 119, 32-43 (2017)
[34] Lucca, G.; Sanz, J. A.; Dimuro, G. P.; Bedregal, B.; Bustince, H.; Mesiar, R., C_F)integrals: A new family of pre-aggregation functions with application to fuzzy rule-based classification systems, Inf. Sci., 435, 94-110 (2018) · Zbl 1440.68293
[35] Ma, Z.; Wu, W.-M., Logical operators on complete lattices, Inf. Sci., 55, 77-97 (1991) · Zbl 0741.03010
[36] Mas, M.; Mayor, G.; Torrens, J., t-Operators and uninorms on a finite totally ordered set, Int. J. Intell. Syst., 14, 909-922 (1999) · Zbl 0948.68173
[37] R. Paiva, E. Palmeira, R. Santiago, B. Bedregal, Lattice-valued overlap and quasi-overlap functions, arXiv preprint (2019), https://arxiv.org/pdf/1902.00133.
[38] Palmeira, E. S.; Bedregal, B.; Bustince, H.; Paternain, D.; Miguel, L. D., Application of two different methods for extending lattice-valued restricted equivalence functions used for constructing similarity measures on L-fuzzy sets, Inf. Sci., 441, 95-112 (2018) · Zbl 1440.03059
[39] Paternain, D.; Bustince, H.; Pagola, M.; Sussner, P.; Kolesárová, A.; Mesiar, R., Capacities and overlap indexes with an application in fuzzy rule-based classification systems, Fuzzy Sets Syst., 305, 70-94 (2016) · Zbl 1368.68298
[40] Qiao, J.; Hu, B. Q., On interval additive generators of interval overlap functions and interval grouping functions, Fuzzy Sets Syst., 323, 19-55 (2017) · Zbl 1376.03052
[41] Qiao, J.; Hu, B. Q., On homogeneous, quasi-homogeneous and pseudo-homogeneous overlap and grouping functions, Fuzzy Sets Syst., 357, 58-90 (2019) · Zbl 1423.03223
[42] Ti, L.; Zhou, H., On (O, N))coimplications derived from overlap functions and fuzzy negations, J. Intell. Fuzzy Syst., 34, 3993-4007 (2018)
[43] Wang, Y.-M.; Liu, H.-W., The modularity condition for overlap and grouping functions, Fuzzy Sets Syst., 372, 97-110 (2019) · Zbl 1423.03234
[44] Wang, Y.-M.; Zhan, H.; Liu, H.-W., Uni-nullnorms on bounded lattices, Fuzzy Sets Syst., 386, 132-144 (2020) · Zbl 1465.03090
[45] Zhang, D., 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.