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 $f$.
Categories: math.GN