We introduce the concept of fuzzy r-minimal preopen set on a fuzzy minimal space. We also introduce the concept of fuzzy r-M precontinuous mapping which is a generalization of fuzzy r-M continuous mapping, and investigate characterization of fuzzy r-M precontinuity. In this paper, we introduce the concept of fuzzy  -minimal preopen sets and study some related properties. We also introduce the concept of fuzzy -M precontinuous mappings, which is a generalization of fuzzy -M continuous mapping. In particular, we investigate some characterization for the fuzzy -M precontinuous mapping in terms of fuzzy -minimal interior and fuzzy -minimal closure operators.

fuzzy r-minimal 공간에서 fuzzy r-  preopen 함수의 개념을 소개하며 fuzzy r- pre-연속 함수, fuzzy r-  preopen 함수와 여러 종류의 fuzzy r-minimal 컴팩트성과의 관계에 대하여 조사한다. AbstractIn this paper, we introduce and study the concept of fuzzy r-  preopen mappings between fuzzy r-minimal spaces.We also investigate the relationships among fuzzy r- precontinuous mappings, fuzzy r-  -preopen mappings and several types of fuzzy r-minimal compactness.Key Words : fuzzy r-minimal preopen, fuzzy r-M continuous, fuzzy r- precontinuous, fuzzy r-minimal precompact, fuzzy r-  -preopen mappings IntorductionThe concept of fuzzy set was introduced by Zadeh [7]. Chang [1] defined fuzzy topological spaces using fuzzy sets. In [2], Chattopadhyay, Hazra and Samanta introduced the smooth topological space which is a generalization of a fuzzy topological space. In [5], we introduced the concept of fuzzy -minimal space which is an extension of the smooth topological space. The concepts of fuzzy -open sets and fuzzy -M continuous mappings are also introduced and studied. In [3], We introduced the concepts of fuzzy -minimal preopen sets and fuzzy -M precontinuous mappings, which are generalizations of fuzzy -minimal open sets and fuzzy -M continuous mappings, respectively. Yoo et al. introduced the concepts of fuzzy -minimal compactness, almost fuzzy -minimal compactness and nearly fuzzy -minimal compactness on fuzzy  -minimal spaces in [6]. In this paper, we introduce and study the concept of fuzzy -  -preopen mapping between fuzzy -minimal spaces. We also investigate the relationships among fuzzy -M precontinuous mappings, fuzzy -  -preopen mappings and several types of fuzzy r-minimal compactness. In particular, in Theorem 3.11, we will show that: If a mapping    →  is fuzzy -M precontinuous and fuzzy -  -preopen on two -FMS's, and If  is a nearly fuzzy -minimal precompact set, then () is a nearly fuzzy -minimal compact set.