Summary: Let denote a locally compact metric space and be a continuous map. In the 1970s, Zadeh presented an extension principle helping us to fuzzify the dynamical system , i.e., to obtain a map for the space of fuzzy sets on . We extend an idea mentioned in [P. Diamond and A. Pokrovskii, Fuzzy Sets Syst. 61, No. 3, 277–283 (1994; Zbl 0827.58037)] to generalize Zadeh’s original extension principle.
In this paper, we study basic properties of so-called -fuzzifications, such as their continuity properties. We also show that, for any -fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems.
Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, the endograph and levelwise topologies.