Continuous, locally Lipschitz and Lipschitz selections for intuitionistic fuzzy sets, with applications to expert voting
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 $D_{\vartheta}$ for a suitable pointwise parameter $\vartheta$. 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 $μ+ν\leq 1$, 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.