Fuzzy closure relations. (English) Zbl 1522.06025
Summary: The concept of closure operator is key in several branches of mathematics. In this paper, closure operators are extended to relational structures, more specifically to fuzzy relations in the framework of complete fuzzy lattices. The core of the work is the search for a suitable definition of (strong) fuzzy closure relation, that is, a fuzzy relation whose relation with fuzzy closure systems is one-to-one. The study of the properties of fuzzy closure systems and fuzzy relations helps narrow down this exploration until an appropriate definition is settled.
MSC:
| 06D72 | Fuzzy lattices (soft algebras) and related topics |
| 06B23 | Complete lattices, completions |
| 06A15 | Galois correspondences, closure operators (in relation to ordered sets) |
References:
| [1] | Bělohlávek, R., Fuzzy closure operators, J. Math. Anal. Appl., 262, 473-489 (2001) · Zbl 0989.54006 |
| [2] | Bělohlávek, R., Fuzzy Relational Systems (2002), Springer · Zbl 1027.06008 |
| [3] | Bělohlávek, R., Concept lattices and order in fuzzy logic, Ann. Pure Appl. Log., 128, 1, 277-298 (2004) · Zbl 1060.03040 |
| [4] | Bělohlávek, R., Pavelka-style fuzzy logic in retrospect and prospect, Fuzzy Sets Syst., 281, 61-72 (2015) · Zbl 1368.03026 |
| [5] | Bělohlávek, R.; De Baets, B.; Outrata, J.; Vychodil, V., Computing the lattice of all fixpoints of a fuzzy closure operator, IEEE Trans. Fuzzy Syst., 18, 3, 546-557 (2010) |
| [6] | Biacino, L.; Gerla, G., Closure systems and L-subalgebras, Inf. Sci., 33, 3, 181-195 (1984) · Zbl 0562.06004 |
| [7] | Bodenhofer, U., A Similarity-Based Generalization of Fuzzy Orderings (01 1999), Johannes-Kepler-Universität Linz |
| [8] | Cabrera, I. P.; Cordero, P.; Muñoz-Velasco, E.; Ojeda-Aciego, M.; De Baets, B., Relational Galois connections between transitive fuzzy digraphs, Math. Methods Appl. Sci., 43, 9, 5673-5680 (2020) · Zbl 1446.06006 |
| [9] | Cabrera, I. P.; Cordero, P.; Muñoz-Velasco, E.; Ojeda-Aciego, M.; De Baets, B., Relational Galois connections between transitive digraphs: characterization and construction, Inf. Sci., 519, 439-450 (2020) · Zbl 1457.05041 |
| [10] | Cabrera, I. P.; Cordero, P.; Muñoz-Velasco, E.; Ojeda-Aciego, M.; De Baets, B., Fuzzy relational Galois connections between fuzzy transitive digraphs, Fuzzy Sets Syst. (2022), submitted for publication · Zbl 1446.06006 |
| [11] | Cordero, P.; Enciso, M.; Mora, A.; Vychodil, V., Parameterized simplification logic I: reasoning with implications and classes of closure operators, Int. J. Gen. Syst., 49, 7, 724-746 (2020) |
| [12] | De Baets, B., Analytical Solution Methods for Fuzzy Relational Equations, 291-340 (2000), Springer US: Springer US Boston, MA · Zbl 0970.03044 |
| [13] | Demirci, M., Fuzzy functions and their applications, J. Math. Anal. Appl., 252, 1, 495-517 (2000) · Zbl 0973.03071 |
| [14] | Elorza, J.; Fuentes-González, R.; Bragard, J.; Burillo, P., On the relation between fuzzy closing morphological operators, fuzzy consequence operators induced by fuzzy preorders and fuzzy closure and co-closure systems, Fuzzy Sets Syst., 218, 73-89 (2013) · Zbl 1350.68273 |
| [15] | Fang, J.; Yue, Y., L-fuzzy closure systems, Fuzzy Sets Syst., 161, 9, 1242-1252 (2010), Foundations of Lattice-Valued Mathematics with Applications to Algebra and Topology · Zbl 1195.54015 |
| [16] | Hájek, P., Metamathematics of Fuzzy Logic, Trends in Logic (2013), Springer Netherlands · Zbl 0937.03030 |
| [17] | Höhle, U.; Blanchard, N., Partial ordering in L-underdeterminate sets, Inf. Sci., 35, 2, 133-144 (1985) · Zbl 0576.06004 |
| [18] | Klawonn, F.; Castro, J., Similarity in fuzzy reasoning, Mathw. Soft Comput., 2, 3, 2 (03 1996) |
| [19] | Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms, vol. 8 (2013), Springer Science & Business Media |
| [20] | Konecny, J.; Krupka, M., Complete relations on fuzzy complete lattices, Fuzzy Sets Syst., 320, 64-80 (2017) · Zbl 1426.06004 |
| [21] | Kuhr, T.; Vychodil, V., Fuzzy logic programming reduced to reasoning with attribute implications, Fuzzy Sets Syst., 262, 1-20 (2015) · Zbl 1360.68311 |
| [22] | Liu, Y.-H.; Lu, L.-X., L-closure operators, L-closure systems and L-closure L-systems on complete L-ordered sets, (2010 Seventh International Conference on Fuzzy Systems and Knowledge Discovery, vol. 1 (2010)), 216-218 |
| [23] | Madrid, N.; Medina, J.; Ojeda-Aciego, M.; Perfilieva, I., Toward the use of fuzzy relations in the definition of mathematical morphology operators, J. Fuzzy Set Valued Anal., 2016, 1, 87-98 (2016) · Zbl 1356.68255 |
| [24] | Moore, E. H., Introduction to a Form of General Analysis, vol. 2 (1910), Yale University Press |
| [25] | Ojeda-Hernández, M.; Cabrera, I. P.; Cordero, P.; Muñoz-Velasco, E., Closure systems as a fuzzy extension of meet-subsemilattices, (Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP) (2021), Atlantis Press), 40-47 |
| [26] | Poelmans, J.; Kuznetsov, S. O.; Ignatov, D. I.; Dedene, G., Formal concept analysis in knowledge processing: a survey on models and techniques, Expert Syst. Appl., 40, 16, 6601-6623 (2013) |
| [27] | Yao, W.; Lu, L. X., Fuzzy Galois connections on fuzzy posets, Math. Log. Q., 55, 105-112 (2009) · Zbl 1172.06001 |
| [28] | Zadeh, L., Fuzzy sets, Inf. Control, 8, 338-353 (1965) · Zbl 0139.24606 |
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