Fréchet fuzzy metric (Q2086086)

The paper deals with the Fréchet distance in a fuzzy metric space in the sense of George and Veeramani. The fuzzy distance between the curves \(\gamma_1, \gamma_2\) is defined as \(M_F (\gamma_1, \gamma_2, t) = \sup \inf M(\gamma_1(\alpha(s)), \gamma_2(s), t)\), where the inf is taken over all \(s\) in the unit interval and the sup is taken over all the homeomorphisms \(\alpha\) of the unit interval. It is shown that \(M_F\) is a metric on the set of all continuous mappings on the universe. Similar results are shown also for the fuzzy metric \(M'(x,y,t) = e^{-\frac{d(x,y)}{t^n}}, n \in N\). A non-monotonic version of the Fréchet distance is considered as well.