Limits of fuzzy retractions of fuzzy hyperspheres and their foldings

Abstract

This article introduces the types of fuzzy retractions of fuzzy hyperspheres $ \tilde{S}^n\subset R^{n+1} $ and the corresponding induced fuzzy retractions of the two fuzzy systems $ \bigcup \overline{\tilde{S}}^n $, $ \bigcup \underline{\tilde{S}}^n $ when the fuzzification is regular or irregular. Also, the fuzzy folding of fuzzy hyperspheres $ \tilde{S}^n $ into itself and the induces two chains of fuzzy folding of $ \bigcup \overline{\tilde{S}}^n $, $ \bigcup \underline{\tilde{S}}^n $. The relations between the fuzzy folding and fuzzy retraction are obtained. The limits of fuzzy retractions and fuzzy folding are achieved.