Transitivity of some uniformities on fuzzy sets

TL;DR

This paper investigates the transitivity properties of various uniformities on the space of normal upper semicontinuous fuzzy sets with compact support, and characterizes when the induced dynamical systems are transitive.

Contribution

It provides a characterization of transitivity for dynamical systems induced by continuous functions on fuzzy set spaces under different uniformities.

Findings

Characterization of transitivity for level-wise uniformity

Characterization of transitivity for Skorokhod uniformity

Characterization of transitivity for sendograph uniformity

Abstract

Given a uniform space $(X, \mathcal{U})$, we denote by $\mathcal{F}(X)$ to the family of all normal upper semicontinuous fuzzy sets $u \colon X \to [0,1]$ with compact support. In this paper, we study transitivity on some uniformities on $\mathcal{F}(X)$: the level-wise uniformity $\mathcal{U}_{\infty}$, the Skorokhod uniformity $\mathcal{U}_{0}$, and the sendograph uniformity $\mathcal{U}_S$. If $f \colon (X, \mathcal{U}) \to (X, \mathcal{U})$ is a continuous function, we mainly characterize when the induced dynamical systems $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{\infty}) \to (\mathcal{F}(X), \mathcal{U}_{\infty})$, $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{0}) \to (\mathcal{F}(X), \mathcal{U}_{0})$ and $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{S}) \to (\mathcal{F}(X), \mathcal{U}_{S})$ are transitive, where $\widehat{f}$ is the Zadeh's extension of $f$.