Continuous, locally Lipschitz and Lipschitz selections for intuitionistic fuzzy sets, with applications to expert voting
Abstract
This paper studies continuous selections in the framework of intuitionistic fuzzy sets. To each intuitionistic fuzzy set A=(μ,ν) on a topological space X we associate the interval-valued map FA(x)=[μ(x),1−ν(x)], whose width is the hesitation degree. Hence, in this setting, a selection of FA is an ordinary fuzzy membership function compatible with the intuitionistic information, and every selection is the membership function of Atanassov's operator Dϑ for a suitable pointwise parameter ϑ. When X is normal and μ, ν are upper semi-continuous, a continuous selection exists by the Katětov--Tong insertion theorem. We show that upper semi-continuity arises naturally in a voting model, in which a panel of experts divides a metric universe into closed regions of acceptance and rejection, and for this model we construct an explicit continuous selection, the consensus selection, illustrated by a non-compact example in the plane. On metric spaces we characterise the intuitionistic fuzzy sets that admit an L-Lipschitz selection through a non-local version of the condition μ+ν≤ 1, and we give the largest and the smallest such selections in closed form. Finally, we prove that the set of continuous selections depends in a $1$-Lipschitz way on the intuitionistic fuzzy set, with respect to the Hausdorff-type distance of Grzegorzewski. We also characterise the intuitionistic fuzzy sets admitting locally Lipschitz selections.
Related
- Existence of Dynamical Low-Rank Approximation for SDEs with Locally Lipschitz Coefficients
- Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data
- Multivariable Extremum Seeking for Locally Lipschitz Objectives
- Continuous solutions to the quaternionic Monge-Ampère equation on locally flat HKT manifolds
- Maximal Monotone Differential Inclusions with Volterra and One-Sided Lipschitz Perturbations under Nonlocal Initial Conditions and Applications
BibTeX
@misc{failla2026continuous,
title = {{Continuous, locally Lipschitz and Lipschitz selections for intuitionistic fuzzy sets, with applications to expert voting}},
author = {Giuseppe Failla and Gianmarco La Rosa and Marco Elio Tabacchi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00857},
eprint = {2611.00857},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00857},
}
RIS
TY - UNPB
TI - Continuous, locally Lipschitz and Lipschitz selections for intuitionistic fuzzy sets, with applications to expert voting
AU - Giuseppe Failla
AU - Gianmarco La Rosa
AU - Marco Elio Tabacchi
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00857
UR - https://arxiv.org/abs/2611.00857
AB - This paper studies continuous selections in the framework of intuitionistic fuzzy sets. To each intuitionistic fuzzy set A=(μ,ν) on a topological space X we associate the interval-valued map F_A(x)=[μ(x),1−ν(x)], whose width is the hesitation degree. Hence, in this setting, a selection of F_A is an ordinary fuzzy membership function compatible with the intuitionistic information, and every selection is the membership function of Atanassov's operator D_ϑ for a suitable pointwise parameter ϑ. When X is normal and μ, ν are upper semi-continuous, a continuous selection exists by the Katětov--Tong insertion theorem. We show that upper semi-continuity arises naturally in a voting model, in which a panel of experts divides a metric universe into closed regions of acceptance and rejection, and for this model we construct an explicit continuous selection, the consensus selection, illustrated by a non-compact example in the plane. On metric spaces we characterise the intuitionistic fuzzy sets that admit an L-Lipschitz selection through a non-local version of the condition μ+ν≤ 1, and we give the largest and the smallest such selections in closed form. Finally, we prove that the set of continuous selections depends in a $1$-Lipschitz way on the intuitionistic fuzzy set, with respect to the Hausdorff-type distance of Grzegorzewski. We also characterise the intuitionistic fuzzy sets admitting locally Lipschitz selections.
ER -