Fuzzy closure structures as formal concepts
Abstract
Galois connections seem to be ubiquitous in mathematics. They have been used to model solutions for both pure and application-oriented problems. Throughout the paper, the general framework is a complete fuzzy lattice over a complete residuated lattice. The existence of three fuzzy Galois connections (two antitone and one isotone) between three specific ordered sets is proved in this paper. The most interesting part is that fuzzy closure systems, fuzzy closure operators and strong fuzzy closure relations are formal concepts of these fuzzy Galois connections.
Funding
FLAIR: Fuzzy, Logic and Algebraic tools for Information Resources
Citation
Please, cite this work as:
[Oje+23] M. Ojeda-Hernández, I. P. Cabrera, P. Cordero, et al. “Fuzzy closure structures as formal concepts”. In: Fuzzy Sets and Systems 463 (2023). Cited by: 6; All Open Access, Green Open Access, Hybrid Gold Open Access. DOI: 10.1016/j.fss.2022.12.014. URL: [https://www.scopus.com/inward/record.uri?eid=2-s2.0-85145982725&doi=10.1016
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Papers citing this work
The following is a non-exhaustive list of papers that cite this work:
[1] R. G. Aragón, J. Medina, and S. Molina-Ruiz. “The Notion of Bond in the Multi-adjoint Concept Lattice Framework”. In: Advances in Artificial Intelligence. Springer Nature Switzerland, 2024, p. 243–253. ISBN: 9783031627996. DOI: 10.1007/978-3-031-62799-6_25. URL: http://dx.doi.org/10.1007/978-3-031-62799-6_25.
[2] R. G. Aragón, J. Medina, and E. Ramírez-Poussa. “Factorizing formal contexts from closures of necessity operators”. In: Computational and Applied Mathematics 43.3 (Mar. 2024). ISSN: 1807-0302. DOI: 10.1007/s40314-024-02590-0. URL: http://dx.doi.org/10.1007/s40314-024-02590-0.
[3] R. G. Aragón, J. Medina, and E. Ramírez-Poussa. “Independent subcontexts and blocks of concept lattices. Definitions and relationships to decompose fuzzy contexts”. In: Fuzzy Sets and Systems 509 (Jun. 2025), p. 109345. ISSN: 0165-0114. DOI: 10.1016/j.fss.2025.109345. URL: http://dx.doi.org/10.1016/j.fss.2025.109345.
[4] M. E. Cornejo, J. Medina, and F. J. Ocaña. “Attribute implications in multi-adjoint concept lattices with hedges”. In: Fuzzy Sets and Systems 479 (Mar. 2024), p. 108854. ISSN: 0165-0114. DOI: 10.1016/j.fss.2023.108854. URL: http://dx.doi.org/10.1016/j.fss.2023.108854.
[5] D. Dubois, J. Medina, and H. Prade. “Extracting attribute implications from a formal context: Unifying the basic approaches”. In: Information Sciences 689 (Jan. 2025), p. 121419. ISSN: 0020-0255. DOI: 10.1016/j.ins.2024.121419. URL: http://dx.doi.org/10.1016/j.ins.2024.121419.
[6] M. Ojeda-Hernández, I. P. Cabrera, P. Cordero, et al. “Fuzzy closure structures as formal concepts II”. In: Fuzzy Sets and Systems 473 (Dec. 2023), p. 108734. ISSN: 0165-0114. DOI: 10.1016/j.fss.2023.108734. URL: http://dx.doi.org/10.1016/j.fss.2023.108734.
[7] M. Ojeda-Hernández, I. P. Cabrera, P. Cordero, et al. “On the Commutative Diagrams Among Galois Connections Involved in Closure Structures”. In: Formal Concept Analysis. Springer Nature Switzerland, 2023, p. 49–63. ISBN: 9783031359491. DOI: 10.1007/978-3-031-35949-1_4. URL: http://dx.doi.org/10.1007/978-3-031-35949-1_4.