Abstract
In this paper, we studied the relations between new types of fuzzy retractions, fuzzy foldings, and fuzzy deformation retractions, on fuzzy fundamental groups of the fuzzy Minkowski space M ˜ 4 . These geometrical transformations are used to give a combinatorial characterization of the fundamental groups of fuzzy submanifolds on M ˜ 4 . Then, the fuzzy fundamental groups of the fuzzy geodesics and the limit fuzzy foldings of M ˜ 4 are presented and obtained. Finally, we proved a sequence of theorems concerning the isomorphism between the fuzzy fundamental group and the fuzzy identity group.
Highlights
Fuzzy sets defined by Zadeh [1] have been widely combined to several mathematical fields, such as complex numbers, topological foldings, algebra, and many more fields
Anthony and Sherwood [17] presented the notion of triangular norm which is used to redefine the fuzzy subgroup
Youssef and Dib [8, 9] announced an innovative approach to define the fuzzy groupoid and fuzzy subgroupoid. is approach appeared in the theory of fuzzy algebra because of the absence of the concepts of the fuzzy universal set and the fuzzy binary operation which were presented by Youssef and Dib in [8, 9]. e main difference between the Rosenfeld’s approach [6]
Summary
Fuzzy sets defined by Zadeh [1] have been widely combined to several mathematical fields, such as complex numbers (see Ramot et al [2, 3]), topological foldings (see [4, 5]), algebra (see Rosenfeld, Dib, Youssef, and El-Ahmady [6,7,8,9,10,11,12]), and many more fields. Is approach appeared in the theory of fuzzy algebra because of the absence of the concepts of the fuzzy universal set and the fuzzy binary operation which were presented by Youssef and Dib in [8, 9]. Abu-Saleem [24] introduced a new type of the fundamental group and studied some types of conditional foldings and unfoldings restricted to the elements of the fuzzy fundamental groups. He presented some theorems and corollaries about the fuzzy fundamental groups of the limit of foldings and the variant and invariant of the fuzzy fundamental group under the folding of the fuzzy manifold into itself. Haçat [25] studied fuzzy H-space and fuzzy H-group and shown that a fuzzy deformation retract of a fuzzy loop space is a fuzzy H-group
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