Research Article On Fuzzy Fundamental Groups and Fuzzy Folding of Fuzzy Minkowski Space

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 com-binatorial 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

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] and that of Youssef and Dib [8,9] is the replacement of the t− norm f with a family of comembership functions f xy : x, y ∈ X . erefore, Dib [9] presented the concept of fuzzy space. Zavadskas et al. [23] applied the idea of Minkowski space with ARAS, TOPSIS, and weighted product methods, and they presented a new model to be applied for numerous problems in which the expert's knowledge is needed to make a proper decision. 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. Also, 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. Later, 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. e background of this paper is considered as a continuation of the above efforts in following the study of fuzzy groups by Rosenfild's [6], and it starts from the definition of isometric folding map of Riemannian manifolds by Robertson [26], who defined this map between two Riemannian manifolds and stated some of its properties such as the continuity property and the property of conservativeness on the length of piecewise geodesic paths.
Depending on this definition of isometric folding maps, El-Ghoul [27] introduced the concept of folding fuzzy graphs and studied its relation with the fuzzy spheres, and then El-Ghoul and Shamara [28] studied the retraction relation between these kinds of folding fuzzy manifolds.
Another achieved progress had been made while studying the folding of fuzzy manifolds through introducing the definitions of isometric folding map into fuzzy retraction, fuzzy foldings, fuzzy deformation, the folding of fuzzy horocycle, and the folding in the fuzzy Lobachevskian space [29,30]. e aim of this paper is to characterize the fuzzy fundamental groups of fuzzy Minkowski space and their isomorphisms, which is to the best of our knowledge not done yet by anyone else and to study new types of fuzzy retractions, fuzzy folding, and fuzzy deformation retract of fuzzy fundamental groups in M 4 . is aim motivated us to do this paper and to prove a sequence of theorems focused on the isomorphisms on foldings on M 4 space. is characterization is very useful because of its wide range of applications, such as the magneto-static atmospheres, the magnetic forces of some manifolds, and so on, which could be found in [5,8,12,[31][32][33]].

Materials and Methods
Our methodology in this paper depends on constructing an isometric folding map on the fuzzy Minkowski space starting from the fuzzy geodesic of the fuzzy Buchdahi space Definition 1 (see [29][30][31]). A fuzzy subset (A, μ) of a fuzzy manifold (M, μ) is called a fuzzy retraction of (M, μ) if there exist a continuous map r: Definition 2 (see [29][30][31]). A fuzzy subset (M, μ) of a fuzzy manifold (M, μ) is called fuzzy deformation retract if there exists a fuzzy retraction r: (M, μ) ⟶ (M, μ) and a fuzzy homotopy ϕ: is the retraction mentioned above.

Results and Discussion
In this section, we present and prove some theorems and results describing the relation between fuzzy fundamental groups and each of fuzzy folding S 1 2 ⊂ M 4 , limit of fuzzy geodesic, minimal fuzzy retraction, and some other results.
Theorem 1 (see [4,11,34,35]). e fuzzy fundamental group of types of fuzzy deformation retracts of M 4 is either isomorphic to Z or its a fuzzy identity group.
Proof. We will show that S which is the fuzzy metric of M 4 . e fuzzy cylindrical coordinates of M 4 are given by , where s 1 , s 2 , s 3 , and s 4 are the constants of integration. Solving the Lagrangian equations, we obtain fuzzy geodesics and retractions in M 4 given as the follows: 2 Advances in Fuzzy Systems a fuzzy hypersphere S e fuzzy deformation retract of M 4 is given by is the open fuzzy Minkowski space M 4 , while the fuzzy and the fuzzy deformation retract of (M 4 − μ i ) onto a fuzzy retraction S 1 2 ⊂ M 4 is given by and the fuzzy deformation retract of (M 4 − μ i ) onto a fuzzy retraction S 2 1 ⊂ M 4 is given by ) is isomorphic to the fuzzy identity group. While if the fuzzy folding ζ: , then the isomorphism between S 1 2 ⊂ M 4 into itself will be defined by , and then this type of fuzzy folding and any fuzzy folding isomorphic to it will induce singularity of S Proof. Consider the fuzzy great sphere S 2 1 of two dimension: ⊂ M 4 be a fuzzy folding; if we define the series of fuzzy folding maps by the following: then , then Lim n⟶∞ m � ((s 1 /1 − i) sin(s 2 /1 − ir (η)), 0, (s 1 /1 − i)cos(s 2 /1 − ir(η)), 0}, so − ( then Lim n⟶∞ m � 0, 0, s 1 /1 − i cos (s 2 /1 − ir(η)), isomorphic to the fuzzy identity group and any fuzzy manifold homeomorphic to M 1 2 .

Advances in Fuzzy Systems
Proof. Let ξ 1 : F n ⟶ F n be a fuzzy folding of F n into itself, we have the following chains: e end of the limits of fuzzy foldings is the zero-dimensional fuzzy manifold which is a fuzzy point, and the fuzzy fundamental group of a fuzzy point is the fuzzy identity group.
So, the minimal fuzzy retraction of the n-dimensional fuzzy manifold F n coincides with the limit of the fuzzy retractions which is the zero-dimensional fuzzy point space F 0 , and the fundamental group of a fuzzy point is the fuzzy identity group.