On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems
Metadatos
Afficher la notice complèteAuteur
Jara Martínez, Pascual; Merino González, Luis Miguel; Navarro Garulo, Gabriel; Santos Aláez, EvángelinaEditorial
Institute of Electrical and Electronics Engineers
Materia
Fuzzy sets Gradual sets Hesitant fuzzy sets
Date
2024-08-20Referencia bibliográfica
P. Jara, L. Merino, G. Navarro and E. Santos, "On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems," in IEEE Access, vol. 12, pp. 111158-111168, 2024, doi: 10.1109/ACCESS.2024.3441940
Patrocinador
Programa Operativo FEDER 2014-2020 and Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía, Spain, under Grant A-FQM-394-UGR20; ‘‘María de Maeztu’’ Excellence Unit IMAG, funded by MCIN/AEI/10.13039/501100011033, under Grant CEX2020-001105-MRésumé
We analyze the connection between two perspectives when defining fuzzy sets: the viewpoint
of mappings and the viewpoint of families of level cuts. This analysis is mathematically supported by
the framework of a categorical adjunction, which serves as a dictionary between these two perspectives.
We prove that hesitant fuzzy sets and gradual sets are strongly related through this connection. This allows
concepts and operations to be transferred from one class to the other, and vice versa. Concretely, as an
application, we provide lattice operations on gradual sets, compatible with Zadeh’s max-min operations
on fuzzy sets, when considering them as families of level cuts. We discuss the well-known representation
theorem for fuzzy sets within this framework, and we show that the representation of fuzzy sets as gradual
sets depends on the chosen embedding of fuzzy sets as hesitant fuzzy sets. Hence, distinct embeddings yield
diverse representations of fuzzy sets as collections of subsets. Furthermore, we extend this methodology to
include other classes of extended fuzzy sets. As a consequence, a representation theorem for interval-valued
fuzzy sets is provided.