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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 205-213

Published online June 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.2.205

© The Korean Institute of Intelligent Systems

Soft Topological Folding on the Product of Soft Manifolds and Its Soft Fundamental Group

Mohammed Abu Saleem

Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt, Jordan

Correspondence to :
Mohammed Abu Saleem (mohammedabusaleem2005@yahoo.com)

Received: March 3, 2023; Revised: May 6, 2023; Accepted: June 9, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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In this study, we analyze the practical applications of soft retraction and soft folding theory on the product of soft manifolds and highlight the role of the soft fundamental group. We propose some novel ideas on the theory of soft manifolds, specifically soft topological folding and soft retraction. We also consider soft topological folding on a finite product of soft manifolds to obtain a new functor on the soft fundamental group. Furthermore, we introduce soft topological folding on the wedge sum of soft manifolds and their induced topological folding on a free product of the soft fundamental group. Based on this framework, we obtain the relationship between a sequence of soft retraction and a sequence of soft topological folding on an n-dimensional soft manifold. Our proposed approach provides some notable insights into the relation between soft folding techniques and topological analysis, as well as the computation of the soft fundamental group.

Keywords: Soft manifold, Soft retraction, Soft folding, Soft topological space, Soft fundamental group

1. Introduction

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Molodtsov [1] developed soft set theory to model complicated problems in fields such as physics, medicine, engineering, biology, sociology, economics, and so forth that cannot be represented with conventional mathematical techniques owing to the different categories of uncertainty provided in these questions. Soft set theory is also related to the subjects of fuzzy and rough sets. Given its suitability for modeling uncertainties in fields including the smoothness of functions, game theory, Riemann integration, and operation research, soft set theory has developed into a valuable tool for data mining, reasoning from data, and decision-making in information systems by using parameterization, particularly when uncertainty is included. As a crucial branch of modern mathematics, topology has many applications in computer science and physics. Topological constructions on soft sets provide more general methods for assessing the similarities and differences between soft set objects in a universe [2–10]. The basic operations considered in soft set theory have been established in the literature [11]. These operations, relationships, and concepts were then quickly developed based on this framework [12–15]; many examples of applications of soft sets to algebraic structures have also been provided [5, 6, 16–18]. The characteristics of soft algebra were then explored further [19, 20]. Two recent studies [21, 22] have discussed a novel approach to soft set theory in decision-making with some applications which can be used to solve a wide variety of problems with uncertainties successfully by applying these decision-making techniques. The authors of [13,23,24] proposed the concepts of soft set relations, soft set functions, and soft bi-continuity and studied related notions including ordering on soft sets, equivalence soft set relations, and partitions. Supra-soft topological spaces have also been introduced, and certain characteristics of these spaces have been elucidated [25]. Al-shami and El-Shafei [26, 27] also analyzed two novel categories of ordered soft separation axioms and partial belonging relations in the context of soft separation axioms. Novel forms of compactness in topological spaces with fuzzy soft properties have also been considered [28]. Many applications related to soft separations and new axioms were also extensively explored [29, 30]. Some corrections to certain outcomes related to soft equality and soft relations were also proposed in further work along these lines [31]. The authors of [32–34] also presented the concept of soft mapping and its application to crisp and soft sets as well as their corresponding properties. Similarly, Al-shami and Kocinac [35] also proposed some new extensions of soft Menger spaces. More types of soft open sets have also been suggested with some practical implications via soft topologies [36–38]. The relation between algebras and topologies with various characterizations has also been explored [39–42].

In the present work, we investigate the behavior of soft manifolds under soft folding to understand how soft topological folding affects the soft fundamental group of the product of soft manifolds. Specifically, we explore various types of soft folding that can be applied to the class of manifolds without compromising their topological properties.

To do so, we introduce some novel definitions for soft manifolds, soft topological folding, and soft retractions and examine their properties. Additionally, the concepts are applied to analyze the soft folding on the soft fundamental group of the wedge sum of soft manifolds. Also, we construct the effect of soft topological folding on a soft manifold Msoft or a finite number of products on soft Riemannian manifolds M1soft×M2soft×…×Mnsoft on the soft fundamental group π1soft (Γ(Msoft)) and π1soft (Γ(M1soft×M2soft×…×Mnsoft)).

2. Preliminaries

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Here, we present some results and concepts from the theory of soft sets and soft topological spaces with algebraic structures [1, 5, 8, 14, 43, 44].

Definition 2.1[1, 5]

Let be an initial universe, a set of parameters, and . The elements of are called the soft sets over . Also, given soft sets over and ,

  • (i) is called a soft subset of if . In this instance, we write . We say that the pair and are soft equal if

    and ; accordingly, we write . Also,

    and are said to be soft disjoint if .

  • (ii) The elements and of are defined by 0(c) = φ and .

  • (iii) The set is called a soft point of if there exists in which ℱ(c) ≠ φ and ℱ (c) = φ for each a ≠ c denoted by cℱ.

Definition 2.2[14]

Suppose and . Then, the Cartesian product of and is a soft set for which is defined as .

Definition 2.3[43]

Suppose that and let be maps on which the following hold.

  • (i) For , the image of under ħh,e denoted as is the soft set in which ,

    G(d)={∪{h(ℱ(c)):c∈e-1(d)},e-1(d)≠φ,φ,e-1(d)=φ.

  • (ii) If , then the inverse image of under ħh,e denoted as ℏh,e-1GD is the soft set for which , and the map is called the parametric map.

Definition 2.4[43]

A collection τ of subsets of is said to be a soft topology on if τ satisfies the following criteria.

  • (i) .

  • (ii) If , then .

  • (iii) If ℱCα∈τ for each α in some index set ▿, then ∪αℱCα∈τ. The soft topological space is given by the triple ( ) and the members of τ are called soft open sets in . If , and x ∈ ℱ(c), the soft set is called the c-soft open-neighborhood (open-nbd).

Definition 2.5[8, 44]

The soft topological space ( ) is called a soft T2-space if for each distinct pair of soft points x1, x2 there exists soft open sets and in which and and .

Definition 2.6[43]

Suppose we have two soft topological spaces, namely ( ) and ( ), and let be a parametric map. A map h from to is considered soft e-continuous at x in if for any given and any e(c)-soft open-nbd of h(x) in ( ) there exists a c-soft open-nbd of x in ( ) for which . If the soft e-continuity of map h holds at every point in , it is considered a soft e-continuous map.

Definition 2.7[5]

Suppose we have a subspace J = [0, 1] of the Euclidean space R and let be soft topological spaces. Additionally, suppose is a parametric map. Then, the map is called a soft path from cℱ to if δ is soft e-continuous and δ(0) ∈ ℱ(c), . Also, the map δ is called a soft loop at the soft point cℱ if .

Definition 2.8[5]

Suppose ( ) is a soft topological space and let cℱ be a soft point in this space. If x0 ∈ ℱ(c), the set π1soft (X,C,x0)={[δ]e:δ is a soft loop at cℱ, δ(0)=δ(1)=x0} is called the soft fundamental group of ( ) at cℱ. For simplicity, we denote this as .

Theorem 2.9[5]

Let cℱ and be soft points in soft topological spaces ( ) and ( ), respectively, with ΔC={(c, c):c∈C} and the parametric map defined by eˋ(c)=(e(c), e(c)), x∈ℱ(c) , and y∈G(c) . Then, π1soft (X×Y,C,(x,y)) and π1soft(X,C,x)×π1soft (Y,C,y) are isomorphic.

3. Soft Topological Folding and Retraction on a Soft Manifold

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We can generalize these concepts to the theory of soft manifolds from the perspective of classical manifold theory and transformation theory. To elaborate on this point, we introduce the ideas of soft manifolds and soft topological folding, which involve soft continuous mappings on soft Riemannian manifolds. By exploring the properties of these concepts, we investigate their algebraic structures.

Definition 3.1

Let (M, τ, ) be a soft topological space where M is an initial universe and is a nonempty set of parameters. A Ckn – dimensional soft manifold Msoft on M is a nonempty soft second countable Hausdorff topological space such that

  • (i) Msoft is the union of soft open subsets ℱUα and each is equipped with a soft homeomorphism Θα, taking ℱUα to be an soft open set in ℱRn. i.e., Θα : ℱUα→Θα (ℱUα) ⊆ ℱRn.

  • (ii) If ℱUα ∩ ℱUβ ≠ φ, then the overlap map ΘβΘα-1:Θα (ℱUα∩ℱUβ)→Θβ (ℱUα∩ℱUβ) is a smooth soft map. Each pair (ℱUα, Θα) is called a soft chart on Msoft and the collection of all soft charts is called a smooth soft atlas on Msoft. The soft space Msoft taken together with atlas is referred to as a smooth soft manifold of dimension n or a soft Ckn–manifold. Moreover, if the soft topological space Msoft satisfies condition (i) only the soft manifold is referred to as a soft topological manifold or simply a soft manifold. Moreover, for a soft smooth manifold Msoft, a Riemannian manifold is a pair (Msoft, g)where Msoft is a smooth manifold and g is a soft Riemannian metric on Msoft.

Example 3.2

A soft Euclidean Rnsoft is a soft n-dimensional manifold and the soft unit n-dimensional sphere Snsoft is a soft n-manifold.

Definition 3.3

A soft geodesic on a soft Riemannian manifold Msoft is a soft parameterized curve with a soft tangent ℱT parallel along the soft path δ.

Definition 3.4

A soft path δ : ℱ[a, b]→ Msoft is a piecwise soft geodesic if for some subdivision a = a0< a1< a2< · · · < ab = b of [a, b], δ | ℱ[a, b] is a soft segment geodesic of Msoft.

Definition 3.5

A soft manifold Msoft is said to be soft arcwise connected if there exists a soft path between any two soft points on the soft manifold Msoft.

Definition 3.6

A soft subset of a soft topological space is called a soft retract of if there exists a soft continuous map (called a soft retraction) for which . A simple representation of a soft retract of a soft space is the center of a soft circle on a soft Mobius strip.

Definition 3.7

Let Msoft and Nsoft be two soft Riemannian manifolds (not necessarily of the same dimension). A map Λ : Msoft→ Nsoft is said to be a soft topological folding of Msoft into Nsoft if for each piecewise soft geodesic δ : Jsoft → Msoft, J = [a, b], and the induced soft path Λ ∘ δ : Jsoft → Nsoft is a piecewise soft geodesic. Also, if Λ : Msoft → Nsoft preserves the length of soft paths, we refer to it as an isometric soft folding of Msoft into Nsoft.

Definition 3.8

Given soft spaces and Ysoft with chosen soft points x0 ∈ mathcal Xsoft and , the soft wedge sum is the quotient of the soft disjoint union acquired by identifying x0 and y0 at a single soft point and denoted by .

Remark 3.9

Let Msoft be a soft m-dimensional manifold. Then, any soft subset of Msoft is an m-dimensional manifold.

Theorem 3.10

If Msoft is a soft m-dimensional manifold and Nsoft is a soft n-dimensional manifold, then the soft product manifold (M × N)soft is a soft (m+ n)-dimensional manifold.

Proof

First, we note that ℱUm×Un is soft homeomorphic to ℱUm+n because for any positive integer s, ℱUs is soft homeomorphic to ℱRs and ℱRm×Rn is soft homeomorphic to ℱRm+n. Now, we observe that every soft point of ℱM×N has a soft neighborhood ℱUm×Un that is soft homeomorphic to ℱUm+n. Hence, the soft manifold ℱUm×Un is a soft (m+n)-dimensional manifold.

Theorem 3.11

If M1soft,M2soft,…,Mnsoft are soft arcwise connected Riemannian manifolds and Λ is a soft topological folding from ⋁i=1nMjsoft into itself, then there exists an induced soft topological folding Λ̄ of *j=1nπ1soft (Mjsoft) into itself that reduces the degree of *j=1nπ1soft (Mj).

Proof

Let Γ:⋁j=1nMjsoft→⋁j=1nMjsoft be a soft topological folding of ⋁j=1Mjsoft into itself. Then, Γ:⋁i=1nMjsoft→⋁i=1nMjsoft has the following forms. If Γ(⋁j=1nMjsoft)=M1soft∨M2soft∨…∨Γ(Mssoft)∨…∨Mnsoft for s = 1, 2, …, n, then Γ¯ (*j=1nπ1soft (Mjsoft))=π1soft(Γ(⋁j=1nMjsoft))≈π1soft (M1soft)*π1soft (M2soft)*…*π1soft (Γ (Mssoft))*…*π1soft (Mnsoft). Given that π1soft (Γ (Mssoft))≤degree (π1soft (Mssoft)), it follows that Λ reduces the degree of *j=1π1soft(Mjsoft). Also, if Γ (⋁j=1nMjsoft)=M1soft∨M2soft∨..∨Γ (Mssoft)∨…∨Γ (Mksoft)∨…∨Mnsoft for k = 1, 2, …, n with s < k, then

Γ¯ (*j=1nπ1soft (Mjsoft))=π1soft (Γ (⋁ij=1nMjsoft))≈π1soft (M1soft)*π1soft (M2soft)*…*π1soft (Γ (Mssoft))*…*π1soft (Γ (Mksoft))*…*π1soft (Mnsoft),

and thus Λ̄ reduces the degree of *j=1nπ1soft (Mjsoft). Moreover, by continuing this process if Γ (⋁j=1nMjsoft)=⋁j=1nΓ(Mjsoft), we obtain Γ¯ (*j=1nπ1soft (Mjsoft))=π1soft (Γ (⋁j=1nMjsoft))≈*j=1nπ1soft (Γ(Mj)). Hence, Λ̄ reduces the degree of *j=1nπ1soft (Mjsoft).

Theorem 3.12

For every k ≤ n, there exists a soft topological folding Λk of ⋁j=1nSj1soft into itself that induces a soft topological folding Λ̄k of *j=1nπ1 (Sj1soft) into itself such that Γ¯k (*j=1nπ1 (Sj1soft)) is a free soft group of rank n − k.

Proof

Let Γ1:⋁j=1nSj1soft→⋁j=1nSj1soft be a soft topological folding such that S11soft∨S21soft∨…∨Γ1(St1soft)∨…∨Sn1soft for t = 1, 2, …, n and a Γ1 (St1)≠St1 topological folding with singularity. Then, there exists an induced soft topological folding Γ¯1:*j=1nπ1soft (Sj1soft)→*j=1nπ1soft (Sj1soft) for which

Γ¯1 (*j=1nπ1soft (Sj1soft))=Γ1 (π1soft(S11soft∨S21soft∨…∨Γ1(St1soft)∨…∨Sn1soft))=π1soft (S11soft)*π1soft (S21soft)*…*π1softΓ1 (St1soft)*…*π1soft (Sn1soft).

Because π1soft (Γ1 (St1soft)) is soft identity group and π1soft(Sj1soft)≈Zsoft, it follows that Γ¯1 (*j=1nπ1soft (Sj1soft)) is a free soft group of rank n − 1. Also, let Γ2:⋁j=1nSj1soft→⋁j=1nSj1soft be a soft topological folding such that Γ2 (⋁j=1nSj1soft)=S11soft∨S21soft∨…∨Γ2 (Ss1soft)∨…∨Γ2 (St1soft)…∨Sn1soft for s, t = 1, 2, …, n, s < t and Γ2 (Ss1soft)≠Ss1soft,Γ2 (St1soft)≠St1soft. Then, the induced soft topological folding Γ¯2:*j=1nπ1soft (Sj1soft)→*j=1nπ1soft (Sj1soft) can be obtained such that Γ¯2 (*j=1nπ1n (Sj1soft)) is a free soft group of rank n − 2. By continuing this process, we obtain the soft topological folding Γn:⋁j=1nSj1soft→⋁j=1nSj1soft such that Γn (⋁j=1nSj1soft)=⋁j=1nΓn (Sj1soft) and Γn (Sj1soft)≠Sj1soft which induces a soft topological folding Γ¯n:*j=1nπ1soft(Sj1soft)→*j=1nπ1soft (Sj1soft) such that Γ¯n (*j=1nπ1soft(Sj1soft)) is a free soft group of rank 0.

Theorem 3.13

If M1soft, M2soft,…Mnsoft are soft arcwise connected Riemannian manifolds and Λ is a topological folding for which Γ(∏j=1nMjsoft)≠∏j=1nΓ(Mjsoft), then

π1soft (limm→∞ (Γm (∏j=1nMjsoft))),∏j=1n(π1soft (limm→∞ (Γm (Mjsoft))))

need not be equal.

Proof

Let us consider M1soft=S1soft, M2soft = S1soft. In this approach, the soft torus can be written as T1soft = S1soft × S1soft. Because limm→∞ (Γm (S1)) is a soft a point, it follows that limm→∞ (Γm (S1soft))×limm→∞ (Γm (S1soft)) is a soft a point and thus π1soft (limm→∞(Γm (S1))×limm→∞(Γm (S1)))=0soft. Also, it follows from λ (S1soft × S1 soft) ≠ Λ (S1soft) × Λ (S1soft) that λ (S1soft × S1 soft) = Λ (S1soft) × S1soft or λ (S1soft × S1soft) = S1soft × ℱ(S1soft) and thus limm→∞ (Γm (S1soft×S1soft))=S1soft. Thus, π1soft(limm→∞Γm(S1soft×S1soft))=π1soft(S1soft)≈Zsoft. Hence, π1soft (limm→∞ (Γm (S1soft×S1soft))), and π1soft (limm→∞(Γm(S1soft)))×π1soft(limm→∞(Γm(S1soft))) are not equal.

Corollary 3.14

If M1soft,M2soft,…,Mnsoft are soft arcwise connected Riemannian manifolds and Λ is a topological folding for which Γ(∏j=1nMjsoft)=∏j=1nΓ(Mjsoft). Then,

π1soft (limm→∞ (Γm (∏j=1nMjsoft)))≈∏j=1n(π1soft (limm→∞ (Γm (Mjsoft)))).
Proof

The proof is obvious.

In the following theorem, we introduce the effect of soft retraction on a finite product of a soft Riemannian manifold and that of the induced soft retraction on their soft fundamental group.

Theorem 3.15

Let M1soft,M2soft,…,Mnsoft be soft Riemannian manifolds. Then, there is a soft retraction from ∏j=1nMjsoft into a soft subset of ∏j=1nMjsoft with the forms

ϒ(∏j=1nMjsoft)=M1soft×M2soft×…×ϒ(Mssoft)×…×Mnsoft,for s=1,2,…,n, or ϒ(∏j=1nMjsoft)=M1soft×M2soft×…×ϒ(Mssoft)×…×ϒ(Mksoft)×…×Mnsoft for s, k=1,2,…,n,s<k,   ⋮ϒ(∏j=1nMjsoft)=ϒ(M1soft)×ϒ(M2soft)×…×ϒ(Mnsoft) induces the following forms.ϒ¯(π1soft (∏j=1nMjsoft))≈π1soft (M1soft)×π1soft (M2soft)×…×π1soft (ϒ(Mssoft))×…×π1soft (Mnsoft), orϒ¯(π1soft (∏j=1nMjsoft))≈π1soft (M1soft)×π1soft (M2soft)×…×π1soft (ϒ(Mssoft))×…×π1soft(ϒ(Mksoft))×…×π1soft (Mnsoft),or⋮ϒ¯(π1soft (∏j=1nMjsoft))=π1soft (ϒ(∏j=1nMjsoft))≈π1soft (ϒ(M1soft))×π1soft (ϒ(M2soft))×…×π1soft (ϒ)(Mnsoft)).

Theorem 3.16

Let Mnsoft be a soft Riemannian manifold of dimension n, Nnsoft be a soft sub Riemannian manifold of Mnsoft, and {Γji,rji, i = 1, 2, …, s, j = 1, 2, …, t} be chains of soft topological foldings and soft retractions. Then, the chain of a commutative diagram on soft Riemannian manifolds and soft sub-Riemannian manifolds induces a chain of a commutative diagram on the soft fundamental groups and the end chain is the soft identity group.

Proof

Consider the chains of commutative diagrams as shown in Table 1.

Using the soft fundamental group as a functor, we obtain the chains of commutative diagrams as shown in Table 2.

Proof

Let ϒ:∏j=1nMjsoft→∏j=1nMjsoft) be a soft retraction with a soft continuous map. In this approach, we obtain the coordinate system of ∏j=1nMjsoft as {(ℱVα11×ℱVα22×…×ℱVαnn),(ℱNα11×ℱNα22×…×ℱNαnn)} where ℱNαjj is soft injective and soft bicontinous mapping from a soft open subset form ℱVαjj⊆Rnjsoft→Mjsoft and {(ℱVαjj,ℱNαjj)} is the soft atlas of Mjsoft for j = 1, 2, …, n. Then, ϒ:∏j=1nMjsoft→ϒ(∏j=1nMjsoft) has the following forms.

If ϒ(∏j=1nMjsoft)=ϒ(ℱVα11×ℱVα22×…×ℱVαnn,ℱNα11×ℱNα22×…×ℱNαnn)=(ℱVα11×ℱVα22×…×ℱVαs-1s-1×ℱVαs+1s+1…×ℱVαnn, ϒ(Vαs,Nαs), ℱNα11×ℱNα22×…×ℱNαs-1s-1×ℱNαs+1s+1…×ℱNαnn)=M1soft×M2soft×…×ϒ(Mssoft)×…×Mnsoft for s = 1, 2, …, n, by ϒ¯(π1soft (∏j=1nMjsoft))=π1soft(ϒ(∏j=1nMj)), and Theorem 2.9 we obtain ϒ¯(π1soft (∏j=1nMjsoft))≈π1soft (M1soft)×π1soft (M2soft)×…×π1soft (ϒ(Mssoft))×…×π1soft (Mnsoft). Also, if ϒ(∏j=1nMjsoft)=ϒ(ℱVα11×ℱVα22×…×ℱVαnn,ℱNα11×ℱNα22×…×ℱNαnn)=(ℱVα11×ℱVα22×…×ℱVαs-1s-1×ℱVαs+1s+1×…×ℱVαk-1k-1×ℱVαk+1k+1×…×ℱVαnn,ϒ(Vαs,Nαs), ϒ(Vαs, Nαs), ϒ(Vαk,Nαk), ℱNα11×ℱNα22×…×ℱNαs-1s-1×ℱNαs+1s+1×…×ℱNαk-1k-1×ℱNαk+1k+1×…×ℱNαnn)=M1soft×M2soft×…×ϒ(Mssoft)×…×ϒ(Mksoft)×…×Mnsoft for s, k = 1, 2, ..., n, s, then ϒ¯(π1 (∏j=1nMjsoft))=π1soft (ϒ(∏j=1nMjsoft))≈π1soft (M1soft)×π1soft (M2soft)×…×π1soft (ϒ(Mssoft))×π1soft(ϒ(Mksoft))×…×π1soft (Mnsoft). Moreover, by continuing this process, if ϒ(∏j=1nMjsoft)=ϒ(ℱVα11×ℱVα22×…×ℱVαnn,ℱNα11×ℱNα22×…×ℱNαnn)=(ϒ(Vα1,Nα1),ϒ(Vα2,Nα2),…,ϒ(Vαn,Nαn))=ϒ(M1soft)×ϒ(M2soft)×…×ϒ(Mnsoft), then ϒ¯(π1soft (∏j=1nMjsoft))=π1soft (ϒ(∏j=1nMjsoft))≈π1soft (ϒ(M1soft))×π1soft (ϒ(M2soft))×…×π1soft (ϒ)Mnsoft)).

We can then derive the soft identity group based on these results and the commutative diagrams.

4. Conclusion

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In this work, we have developed several concepts and tools for use in soft mathematics, including soft manifolds, retraction, folding, and topological spaces, as well as soft fundamental groups. These concepts and tools have been applied in terms of topology and geometry. Furthermore, we have derived a new functor from the soft topological foldings of certain operations on soft manifolds. We have further explored the relationship between a chain of soft retractions and a chain of soft topological folding on an n-dimensional soft manifold.

Tables

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Table. 1.

Table 1. The chains of commutative diagrams.


Table. 2.

Table 2. The chains of commutative diagrams using the soft fundamental group.


References

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Biography

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Mohammed Abu Saleem is an associate professor in the Department of Mathematics at the Faculty of Science, Al-Balqa Applied University, located in Jordan. E-mail: mohammedabusaleem2005@yahoo.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 205-213

Published online June 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.2.205

Copyright © The Korean Institute of Intelligent Systems.

Soft Topological Folding on the Product of Soft Manifolds and Its Soft Fundamental Group

Mohammed Abu Saleem

Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt, Jordan

Correspondence to:Mohammed Abu Saleem (mohammedabusaleem2005@yahoo.com)

Received: March 3, 2023; Revised: May 6, 2023; Accepted: June 9, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this study, we analyze the practical applications of soft retraction and soft folding theory on the product of soft manifolds and highlight the role of the soft fundamental group. We propose some novel ideas on the theory of soft manifolds, specifically soft topological folding and soft retraction. We also consider soft topological folding on a finite product of soft manifolds to obtain a new functor on the soft fundamental group. Furthermore, we introduce soft topological folding on the wedge sum of soft manifolds and their induced topological folding on a free product of the soft fundamental group. Based on this framework, we obtain the relationship between a sequence of soft retraction and a sequence of soft topological folding on an n-dimensional soft manifold. Our proposed approach provides some notable insights into the relation between soft folding techniques and topological analysis, as well as the computation of the soft fundamental group.

Keywords: Soft manifold, Soft retraction, Soft folding, Soft topological space, Soft fundamental group

1. Introduction

Molodtsov [1] developed soft set theory to model complicated problems in fields such as physics, medicine, engineering, biology, sociology, economics, and so forth that cannot be represented with conventional mathematical techniques owing to the different categories of uncertainty provided in these questions. Soft set theory is also related to the subjects of fuzzy and rough sets. Given its suitability for modeling uncertainties in fields including the smoothness of functions, game theory, Riemann integration, and operation research, soft set theory has developed into a valuable tool for data mining, reasoning from data, and decision-making in information systems by using parameterization, particularly when uncertainty is included. As a crucial branch of modern mathematics, topology has many applications in computer science and physics. Topological constructions on soft sets provide more general methods for assessing the similarities and differences between soft set objects in a universe [2–10]. The basic operations considered in soft set theory have been established in the literature [11]. These operations, relationships, and concepts were then quickly developed based on this framework [12–15]; many examples of applications of soft sets to algebraic structures have also been provided [5, 6, 16–18]. The characteristics of soft algebra were then explored further [19, 20]. Two recent studies [21, 22] have discussed a novel approach to soft set theory in decision-making with some applications which can be used to solve a wide variety of problems with uncertainties successfully by applying these decision-making techniques. The authors of [13,23,24] proposed the concepts of soft set relations, soft set functions, and soft bi-continuity and studied related notions including ordering on soft sets, equivalence soft set relations, and partitions. Supra-soft topological spaces have also been introduced, and certain characteristics of these spaces have been elucidated [25]. Al-shami and El-Shafei [26, 27] also analyzed two novel categories of ordered soft separation axioms and partial belonging relations in the context of soft separation axioms. Novel forms of compactness in topological spaces with fuzzy soft properties have also been considered [28]. Many applications related to soft separations and new axioms were also extensively explored [29, 30]. Some corrections to certain outcomes related to soft equality and soft relations were also proposed in further work along these lines [31]. The authors of [32–34] also presented the concept of soft mapping and its application to crisp and soft sets as well as their corresponding properties. Similarly, Al-shami and Kocinac [35] also proposed some new extensions of soft Menger spaces. More types of soft open sets have also been suggested with some practical implications via soft topologies [36–38]. The relation between algebras and topologies with various characterizations has also been explored [39–42].

In the present work, we investigate the behavior of soft manifolds under soft folding to understand how soft topological folding affects the soft fundamental group of the product of soft manifolds. Specifically, we explore various types of soft folding that can be applied to the class of manifolds without compromising their topological properties.

To do so, we introduce some novel definitions for soft manifolds, soft topological folding, and soft retractions and examine their properties. Additionally, the concepts are applied to analyze the soft folding on the soft fundamental group of the wedge sum of soft manifolds. Also, we construct the effect of soft topological folding on a soft manifold Msoft or a finite number of products on soft Riemannian manifolds M1soft×M2soft×…×Mnsoft on the soft fundamental group π1soft (Γ(Msoft)) and π1soft (Γ(M1soft×M2soft×…×Mnsoft)).

2. Preliminaries

Here, we present some results and concepts from the theory of soft sets and soft topological spaces with algebraic structures [1, 5, 8, 14, 43, 44].

Definition 2.1[1, 5]

Let be an initial universe, a set of parameters, and . The elements of are called the soft sets over . Also, given soft sets over and ,

  • (i) is called a soft subset of if . In this instance, we write . We say that the pair and are soft equal if

    and ; accordingly, we write . Also,

    and are said to be soft disjoint if .

  • (ii) The elements and of are defined by 0(c) = φ and .

  • (iii) The set is called a soft point of if there exists in which ℱ(c) ≠ φ and ℱ (c) = φ for each a ≠ c denoted by cℱ.

Definition 2.2[14]

Suppose and . Then, the Cartesian product of and is a soft set for which is defined as .

Definition 2.3[43]

Suppose that and let be maps on which the following hold.

  • (i) For , the image of under ħh,e denoted as is the soft set in which ,

    G(d)={∪{h(ℱ(c)):c∈e-1(d)},e-1(d)≠φ,φ,e-1(d)=φ.

  • (ii) If , then the inverse image of under ħh,e denoted as ℏh,e-1GD is the soft set for which , and the map is called the parametric map.

Definition 2.4[43]

A collection τ of subsets of is said to be a soft topology on if τ satisfies the following criteria.

  • (i) .

  • (ii) If , then .

  • (iii) If ℱCα∈τ for each α in some index set ▿, then ∪αℱCα∈τ. The soft topological space is given by the triple ( ) and the members of τ are called soft open sets in . If , and x ∈ ℱ(c), the soft set is called the c-soft open-neighborhood (open-nbd).

Definition 2.5[8, 44]

The soft topological space ( ) is called a soft T2-space if for each distinct pair of soft points x1, x2 there exists soft open sets and in which and and .

Definition 2.6[43]

Suppose we have two soft topological spaces, namely ( ) and ( ), and let be a parametric map. A map h from to is considered soft e-continuous at x in if for any given and any e(c)-soft open-nbd of h(x) in ( ) there exists a c-soft open-nbd of x in ( ) for which . If the soft e-continuity of map h holds at every point in , it is considered a soft e-continuous map.

Definition 2.7[5]

Suppose we have a subspace J = [0, 1] of the Euclidean space R and let be soft topological spaces. Additionally, suppose is a parametric map. Then, the map is called a soft path from cℱ to if δ is soft e-continuous and δ(0) ∈ ℱ(c), . Also, the map δ is called a soft loop at the soft point cℱ if .

Definition 2.8[5]

Suppose ( ) is a soft topological space and let cℱ be a soft point in this space. If x0 ∈ ℱ(c), the set π1soft (X,C,x0)={[δ]e:δ is a soft loop at cℱ, δ(0)=δ(1)=x0} is called the soft fundamental group of ( ) at cℱ. For simplicity, we denote this as .

Theorem 2.9[5]

Let cℱ and be soft points in soft topological spaces ( ) and ( ), respectively, with ΔC={(c, c):c∈C} and the parametric map defined by eˋ(c)=(e(c), e(c)), x∈ℱ(c) , and y∈G(c) . Then, π1soft (X×Y,C,(x,y)) and π1soft(X,C,x)×π1soft (Y,C,y) are isomorphic.

3. Soft Topological Folding and Retraction on a Soft Manifold

We can generalize these concepts to the theory of soft manifolds from the perspective of classical manifold theory and transformation theory. To elaborate on this point, we introduce the ideas of soft manifolds and soft topological folding, which involve soft continuous mappings on soft Riemannian manifolds. By exploring the properties of these concepts, we investigate their algebraic structures.

Definition 3.1

Let (M, τ, ) be a soft topological space where M is an initial universe and is a nonempty set of parameters. A Ckn – dimensional soft manifold Msoft on M is a nonempty soft second countable Hausdorff topological space such that

  • (i) Msoft is the union of soft open subsets ℱUα and each is equipped with a soft homeomorphism Θα, taking ℱUα to be an soft open set in ℱRn. i.e., Θα : ℱUα→Θα (ℱUα) ⊆ ℱRn.

  • (ii) If ℱUα ∩ ℱUβ ≠ φ, then the overlap map ΘβΘα-1:Θα (ℱUα∩ℱUβ)→Θβ (ℱUα∩ℱUβ) is a smooth soft map. Each pair (ℱUα, Θα) is called a soft chart on Msoft and the collection of all soft charts is called a smooth soft atlas on Msoft. The soft space Msoft taken together with atlas is referred to as a smooth soft manifold of dimension n or a soft Ckn–manifold. Moreover, if the soft topological space Msoft satisfies condition (i) only the soft manifold is referred to as a soft topological manifold or simply a soft manifold. Moreover, for a soft smooth manifold Msoft, a Riemannian manifold is a pair (Msoft, g)where Msoft is a smooth manifold and g is a soft Riemannian metric on Msoft.

Example 3.2

A soft Euclidean Rnsoft is a soft n-dimensional manifold and the soft unit n-dimensional sphere Snsoft is a soft n-manifold.

Definition 3.3

A soft geodesic on a soft Riemannian manifold Msoft is a soft parameterized curve with a soft tangent ℱT parallel along the soft path δ.

Definition 3.4

A soft path δ : ℱ[a, b]→ Msoft is a piecwise soft geodesic if for some subdivision a = a0< a1< a2< · · · < ab = b of [a, b], δ | ℱ[a, b] is a soft segment geodesic of Msoft.

Definition 3.5

A soft manifold Msoft is said to be soft arcwise connected if there exists a soft path between any two soft points on the soft manifold Msoft.

Definition 3.6

A soft subset of a soft topological space is called a soft retract of if there exists a soft continuous map (called a soft retraction) for which . A simple representation of a soft retract of a soft space is the center of a soft circle on a soft Mobius strip.

Definition 3.7

Let Msoft and Nsoft be two soft Riemannian manifolds (not necessarily of the same dimension). A map Λ : Msoft→ Nsoft is said to be a soft topological folding of Msoft into Nsoft if for each piecewise soft geodesic δ : Jsoft → Msoft, J = [a, b], and the induced soft path Λ ∘ δ : Jsoft → Nsoft is a piecewise soft geodesic. Also, if Λ : Msoft → Nsoft preserves the length of soft paths, we refer to it as an isometric soft folding of Msoft into Nsoft.

Definition 3.8

Given soft spaces and Ysoft with chosen soft points x0 ∈ mathcal Xsoft and , the soft wedge sum is the quotient of the soft disjoint union acquired by identifying x0 and y0 at a single soft point and denoted by .

Remark 3.9

Let Msoft be a soft m-dimensional manifold. Then, any soft subset of Msoft is an m-dimensional manifold.

Theorem 3.10

If Msoft is a soft m-dimensional manifold and Nsoft is a soft n-dimensional manifold, then the soft product manifold (M × N)soft is a soft (m+ n)-dimensional manifold.

Proof

First, we note that ℱUm×Un is soft homeomorphic to ℱUm+n because for any positive integer s, ℱUs is soft homeomorphic to ℱRs and ℱRm×Rn is soft homeomorphic to ℱRm+n. Now, we observe that every soft point of ℱM×N has a soft neighborhood ℱUm×Un that is soft homeomorphic to ℱUm+n. Hence, the soft manifold ℱUm×Un is a soft (m+n)-dimensional manifold.

Theorem 3.11

If M1soft,M2soft,…,Mnsoft are soft arcwise connected Riemannian manifolds and Λ is a soft topological folding from ⋁i=1nMjsoft into itself, then there exists an induced soft topological folding Λ̄ of *j=1nπ1soft (Mjsoft) into itself that reduces the degree of *j=1nπ1soft (Mj).

Proof

Let Γ:⋁j=1nMjsoft→⋁j=1nMjsoft be a soft topological folding of ⋁j=1Mjsoft into itself. Then, Γ:⋁i=1nMjsoft→⋁i=1nMjsoft has the following forms. If Γ(⋁j=1nMjsoft)=M1soft∨M2soft∨…∨Γ(Mssoft)∨…∨Mnsoft for s = 1, 2, …, n, then Γ¯ (*j=1nπ1soft (Mjsoft))=π1soft(Γ(⋁j=1nMjsoft))≈π1soft (M1soft)*π1soft (M2soft)*…*π1soft (Γ (Mssoft))*…*π1soft (Mnsoft). Given that π1soft (Γ (Mssoft))≤degree (π1soft (Mssoft)), it follows that Λ reduces the degree of *j=1π1soft(Mjsoft). Also, if Γ (⋁j=1nMjsoft)=M1soft∨M2soft∨..∨Γ (Mssoft)∨…∨Γ (Mksoft)∨…∨Mnsoft for k = 1, 2, …, n with s < k, then

Γ¯ (*j=1nπ1soft (Mjsoft))=π1soft (Γ (⋁ij=1nMjsoft))≈π1soft (M1soft)*π1soft (M2soft)*…*π1soft (Γ (Mssoft))*…*π1soft (Γ (Mksoft))*…*π1soft (Mnsoft),

and thus Λ̄ reduces the degree of *j=1nπ1soft (Mjsoft). Moreover, by continuing this process if Γ (⋁j=1nMjsoft)=⋁j=1nΓ(Mjsoft), we obtain Γ¯ (*j=1nπ1soft (Mjsoft))=π1soft (Γ (⋁j=1nMjsoft))≈*j=1nπ1soft (Γ(Mj)). Hence, Λ̄ reduces the degree of *j=1nπ1soft (Mjsoft).

Theorem 3.12

For every k ≤ n, there exists a soft topological folding Λk of ⋁j=1nSj1soft into itself that induces a soft topological folding Λ̄k of *j=1nπ1 (Sj1soft) into itself such that Γ¯k (*j=1nπ1 (Sj1soft)) is a free soft group of rank n − k.

Proof

Let Γ1:⋁j=1nSj1soft→⋁j=1nSj1soft be a soft topological folding such that S11soft∨S21soft∨…∨Γ1(St1soft)∨…∨Sn1soft for t = 1, 2, …, n and a Γ1 (St1)≠St1 topological folding with singularity. Then, there exists an induced soft topological folding Γ¯1:*j=1nπ1soft (Sj1soft)→*j=1nπ1soft (Sj1soft) for which

Γ¯1 (*j=1nπ1soft (Sj1soft))=Γ1 (π1soft(S11soft∨S21soft∨…∨Γ1(St1soft)∨…∨Sn1soft))=π1soft (S11soft)*π1soft (S21soft)*…*π1softΓ1 (St1soft)*…*π1soft (Sn1soft).

Because π1soft (Γ1 (St1soft)) is soft identity group and π1soft(Sj1soft)≈Zsoft, it follows that Γ¯1 (*j=1nπ1soft (Sj1soft)) is a free soft group of rank n − 1. Also, let Γ2:⋁j=1nSj1soft→⋁j=1nSj1soft be a soft topological folding such that Γ2 (⋁j=1nSj1soft)=S11soft∨S21soft∨…∨Γ2 (Ss1soft)∨…∨Γ2 (St1soft)…∨Sn1soft for s, t = 1, 2, …, n, s < t and Γ2 (Ss1soft)≠Ss1soft,Γ2 (St1soft)≠St1soft. Then, the induced soft topological folding Γ¯2:*j=1nπ1soft (Sj1soft)→*j=1nπ1soft (Sj1soft) can be obtained such that Γ¯2 (*j=1nπ1n (Sj1soft)) is a free soft group of rank n − 2. By continuing this process, we obtain the soft topological folding Γn:⋁j=1nSj1soft→⋁j=1nSj1soft such that Γn (⋁j=1nSj1soft)=⋁j=1nΓn (Sj1soft) and Γn (Sj1soft)≠Sj1soft which induces a soft topological folding Γ¯n:*j=1nπ1soft(Sj1soft)→*j=1nπ1soft (Sj1soft) such that Γ¯n (*j=1nπ1soft(Sj1soft)) is a free soft group of rank 0.

Theorem 3.13

If M1soft, M2soft,…Mnsoft are soft arcwise connected Riemannian manifolds and Λ is a topological folding for which Γ(∏j=1nMjsoft)≠∏j=1nΓ(Mjsoft), then

π1soft (limm→∞ (Γm (∏j=1nMjsoft))),∏j=1n(π1soft (limm→∞ (Γm (Mjsoft))))

need not be equal.

Proof

Let us consider M1soft=S1soft, M2soft = S1soft. In this approach, the soft torus can be written as T1soft = S1soft × S1soft. Because limm→∞ (Γm (S1)) is a soft a point, it follows that limm→∞ (Γm (S1soft))×limm→∞ (Γm (S1soft)) is a soft a point and thus π1soft (limm→∞(Γm (S1))×limm→∞(Γm (S1)))=0soft. Also, it follows from λ (S1soft × S1 soft) ≠ Λ (S1soft) × Λ (S1soft) that λ (S1soft × S1 soft) = Λ (S1soft) × S1soft or λ (S1soft × S1soft) = S1soft × ℱ(S1soft) and thus limm→∞ (Γm (S1soft×S1soft))=S1soft. Thus, π1soft(limm→∞Γm(S1soft×S1soft))=π1soft(S1soft)≈Zsoft. Hence, π1soft (limm→∞ (Γm (S1soft×S1soft))), and π1soft (limm→∞(Γm(S1soft)))×π1soft(limm→∞(Γm(S1soft))) are not equal.

Corollary 3.14

If M1soft,M2soft,…,Mnsoft are soft arcwise connected Riemannian manifolds and Λ is a topological folding for which Γ(∏j=1nMjsoft)=∏j=1nΓ(Mjsoft). Then,

π1soft (limm→∞ (Γm (∏j=1nMjsoft)))≈∏j=1n(π1soft (limm→∞ (Γm (Mjsoft)))).
Proof

The proof is obvious.

In the following theorem, we introduce the effect of soft retraction on a finite product of a soft Riemannian manifold and that of the induced soft retraction on their soft fundamental group.

Theorem 3.15

Let M1soft,M2soft,…,Mnsoft be soft Riemannian manifolds. Then, there is a soft retraction from ∏j=1nMjsoft into a soft subset of ∏j=1nMjsoft with the forms

ϒ(∏j=1nMjsoft)=M1soft×M2soft×…×ϒ(Mssoft)×…×Mnsoft,for s=1,2,…,n, or ϒ(∏j=1nMjsoft)=M1soft×M2soft×…×ϒ(Mssoft)×…×ϒ(Mksoft)×…×Mnsoft for s, k=1,2,…,n,s<k,   ⋮ϒ(∏j=1nMjsoft)=ϒ(M1soft)×ϒ(M2soft)×…×ϒ(Mnsoft) induces the following forms.ϒ¯(π1soft (∏j=1nMjsoft))≈π1soft (M1soft)×π1soft (M2soft)×…×π1soft (ϒ(Mssoft))×…×π1soft (Mnsoft), orϒ¯(π1soft (∏j=1nMjsoft))≈π1soft (M1soft)×π1soft (M2soft)×…×π1soft (ϒ(Mssoft))×…×π1soft(ϒ(Mksoft))×…×π1soft (Mnsoft),or⋮ϒ¯(π1soft (∏j=1nMjsoft))=π1soft (ϒ(∏j=1nMjsoft))≈π1soft (ϒ(M1soft))×π1soft (ϒ(M2soft))×…×π1soft (ϒ)(Mnsoft)).

Theorem 3.16

Let Mnsoft be a soft Riemannian manifold of dimension n, Nnsoft be a soft sub Riemannian manifold of Mnsoft, and {Γji,rji, i = 1, 2, …, s, j = 1, 2, …, t} be chains of soft topological foldings and soft retractions. Then, the chain of a commutative diagram on soft Riemannian manifolds and soft sub-Riemannian manifolds induces a chain of a commutative diagram on the soft fundamental groups and the end chain is the soft identity group.

Proof

Consider the chains of commutative diagrams as shown in Table 1.

Using the soft fundamental group as a functor, we obtain the chains of commutative diagrams as shown in Table 2.

Proof

Let ϒ:∏j=1nMjsoft→∏j=1nMjsoft) be a soft retraction with a soft continuous map. In this approach, we obtain the coordinate system of ∏j=1nMjsoft as {(ℱVα11×ℱVα22×…×ℱVαnn),(ℱNα11×ℱNα22×…×ℱNαnn)} where ℱNαjj is soft injective and soft bicontinous mapping from a soft open subset form ℱVαjj⊆Rnjsoft→Mjsoft and {(ℱVαjj,ℱNαjj)} is the soft atlas of Mjsoft for j = 1, 2, …, n. Then, ϒ:∏j=1nMjsoft→ϒ(∏j=1nMjsoft) has the following forms.

If ϒ(∏j=1nMjsoft)=ϒ(ℱVα11×ℱVα22×…×ℱVαnn,ℱNα11×ℱNα22×…×ℱNαnn)=(ℱVα11×ℱVα22×…×ℱVαs-1s-1×ℱVαs+1s+1…×ℱVαnn, ϒ(Vαs,Nαs), ℱNα11×ℱNα22×…×ℱNαs-1s-1×ℱNαs+1s+1…×ℱNαnn)=M1soft×M2soft×…×ϒ(Mssoft)×…×Mnsoft for s = 1, 2, …, n, by ϒ¯(π1soft (∏j=1nMjsoft))=π1soft(ϒ(∏j=1nMj)), and Theorem 2.9 we obtain ϒ¯(π1soft (∏j=1nMjsoft))≈π1soft (M1soft)×π1soft (M2soft)×…×π1soft (ϒ(Mssoft))×…×π1soft (Mnsoft). Also, if ϒ(∏j=1nMjsoft)=ϒ(ℱVα11×ℱVα22×…×ℱVαnn,ℱNα11×ℱNα22×…×ℱNαnn)=(ℱVα11×ℱVα22×…×ℱVαs-1s-1×ℱVαs+1s+1×…×ℱVαk-1k-1×ℱVαk+1k+1×…×ℱVαnn,ϒ(Vαs,Nαs), ϒ(Vαs, Nαs), ϒ(Vαk,Nαk), ℱNα11×ℱNα22×…×ℱNαs-1s-1×ℱNαs+1s+1×…×ℱNαk-1k-1×ℱNαk+1k+1×…×ℱNαnn)=M1soft×M2soft×…×ϒ(Mssoft)×…×ϒ(Mksoft)×…×Mnsoft for s, k = 1, 2, ..., n, s, then ϒ¯(π1 (∏j=1nMjsoft))=π1soft (ϒ(∏j=1nMjsoft))≈π1soft (M1soft)×π1soft (M2soft)×…×π1soft (ϒ(Mssoft))×π1soft(ϒ(Mksoft))×…×π1soft (Mnsoft). Moreover, by continuing this process, if ϒ(∏j=1nMjsoft)=ϒ(ℱVα11×ℱVα22×…×ℱVαnn,ℱNα11×ℱNα22×…×ℱNαnn)=(ϒ(Vα1,Nα1),ϒ(Vα2,Nα2),…,ϒ(Vαn,Nαn))=ϒ(M1soft)×ϒ(M2soft)×…×ϒ(Mnsoft), then ϒ¯(π1soft (∏j=1nMjsoft))=π1soft (ϒ(∏j=1nMjsoft))≈π1soft (ϒ(M1soft))×π1soft (ϒ(M2soft))×…×π1soft (ϒ)Mnsoft)).

We can then derive the soft identity group based on these results and the commutative diagrams.

4. Conclusion

In this work, we have developed several concepts and tools for use in soft mathematics, including soft manifolds, retraction, folding, and topological spaces, as well as soft fundamental groups. These concepts and tools have been applied in terms of topology and geometry. Furthermore, we have derived a new functor from the soft topological foldings of certain operations on soft manifolds. We have further explored the relationship between a chain of soft retractions and a chain of soft topological folding on an n-dimensional soft manifold.

Table 1 . The chains of commutative diagrams.


Table 2 . The chains of commutative diagrams using the soft fundamental group.


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