- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in Fuzzy Systems

Volume 2013 (2013), Article ID 342805, 7 pages

http://dx.doi.org/10.1155/2013/342805

## Fuzzy Retractions of Fuzzy Open Flat Robertson-Walker Space

^{1}Mathematics Department, Faculty of Science, Taibah University, Madinah, Saudi Arabia^{2}Mathematics Department, Faculty of Science, Tanta University, Tanta 31527, Egypt

Received 24 April 2012; Accepted 28 August 2012

Academic Editor: Ching-Hung Lee

Copyright © 2013 A. E. El-Ahmady and A. S. Al-Luhaybi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Our aim in the present paper is to introduce and study new types of fuzzy retractions of fuzzy open flat Robertson-Walker model. New types of the fuzzy deformation retracts of model are obtained. The relations between the fuzzy foldings and the fuzzy deformation retracts of model are deduced. Types of fuzzy minimal retractions are also presented. New types of homotopy maps are deduced. New types of conditional fuzzy folding are presented. Some commutative diagrams are obtained.

#### 1. Introduction and Background

Robertson-Walker space represents one of the most intriguing and emblematic discoveries in the history of geometry. Although if it were introduced for a purely geometrical purpose, they came into prominence in many branches of mathematics and physics. This association with applied science and geometry generated synergistic effect: applied science gave relevance to Robertson-Walker space and Robertson-Walker space allowed formalizing practical problems [1–6]. As is well known, the theory of retractions is always one of interesting topics in Euclidian and Non-Euclidian spaces and it has been investigated from the various viewpoints by many branches of topology and differential geometry [7–11]. There are many diverse applications of certain phenomena for which it is impossible to get relevant data. It may not be possible to measure essential parameters of a process such as the temperature inside molten glass or the homogeneity of a mixture inside some tanks. The required measurement scale may not exist at all, such as in the case of evaluation of offensive smells, evaluating the taste of foods or medical diagnoses by touching [7, 8, 12–18]. The aim of the present paper is to describe the above phenomena geometrically, specifically concerned with the study of the new types of fuzzy retractions, fuzzy deformation retracts, and fuzzy folding of fuzzy open flat Robertson*-*Walker model. A fuzzy manifold is manifold which has a physical character. This character is represented by the density function , where [7, 8, 12].

A fuzzy subset of a fuzzy manifold is called a fuzzy retraction of if there exist a continuous map such that [7, 8, 12].

A fuzzy subset of a fuzzy manifold is called a fuzzy deformation retract if there exists a fuzzy retraction and a fuzzy homotopy [7, 13, 14] such that, where is the retraction mentioned above.

Topological folding of fuzzy open flat Robertson-Walker space model [7, 8]. A map is said to be an isometric folding of model into itself if and only if for any piecewise fuzzy geodesic path the induced path is a piecewise fuzzy geodesic and of the same length as , where . If does not preserve lengths, then is a topological folding of fuzzy Robertson-Walker space model [12–14].

The fuzzy folding of model is a folding such that and any belong to the upper hypermanifolds down such that for every corresponding points, that is, [15]. See Figure 1.

#### 2. Main Results

Theorem 1. *The fuzzy retractions of model are the fuzzy unit hyperboloid, fuzzy hyperbolic, fuzzy hypersphere, fuzzy circle, and fuzzy minimal manifolds.*

*Proof. *Consider the model with metric
The coordinate of model is
where the ranges are,, and .

Now, we use Lagrangian equations:
To find a fuzzy geodesic which is a fuzzy subset of model, since

then the Lagrangian equations for model are
From (7) we obtain = constant, say , if , we obtain the following cases.

If initially equal or and hence we obtain the following fuzzy geodesics of fuzzy unit hyperboloid , , and , respectively. Also, if , hence we obtain the following coordinates of model given by
which is a fuzzy hyperbolic , which is a fuzzy geodesic retraction. Now, If or and hence we get the fuzzy unit hyperboloid retractions , and, in model, respectively. Also, in a special case if or and hence we get the fuzzy hyperbolic geodesics ,, and in model, respectively. In a special case if , hence we get the coordinates of model represented by
which is a fuzzy sphere , which is a fuzzy geodesic retraction. Also, If , and we obtain the fuzzy retraction, , which is a fuzzy circle . Again, If, we get the following minimal fuzzy geodesic in model.

In what follows, we present some cases of fuzzy deformation retracts of model. The fuzzy deformation retract of model is , where is the closed interval , present as
The fuzzy deformation retract of model into the fuzzy minimal geodesic is
where
The fuzzy deformation retract of model into the fuzzy hyperboloid is

Now, we are going to discuss the fuzzy folding of model. Let , where
An isometric fuzzy folding of model into itself may be defined by
The fuzzy deformation retract of the fuzzy folded () model into the fuzzy folded geodesic () is
with
The fuzzy deformation retract of the fuzzy folded () model into the fuzzy folded geodesic () is

Then, the following theorem has been proved.

Theorem 2. *Under the defined fuzzy folding and any fuzzy folding homeomorphic to this type of fuzzy folding, the fuzzy deformation retract of the fuzzy folded model into the fuzzy folded geodesics is the same as the fuzzy deformation retract of model into the fuzzy geodesics.*

* Proof. *
Now, let the fuzzy folding be defined by , where
The isometric fuzzy folded model is
The fuzzy deformation retract of the fuzzy folded model into the fuzzy folded geodesic is
The fuzzy deformation retract of the fuzzy folded model into the fuzzy folded geodesic is

Then, the following theorem has been proved.

Theorem 3. * Under the defined fuzzy folding and any fuzzy folding homeomorphic to this type of fuzzy folding, the fuzzy deformation retract of the fuzzy folded model into the fuzzy folded geodesics is different from the fuzzy deformation retract of model into the fuzzy geodesics.*

Theorem 4. *Let be a fuzzy hyperboloid in model which is homeomorphic to , and a fuzzy retraction. Then, there is an induced fuzzy retraction such that the following diagram is commutative:*(24)

*Proof. *Under the condition, then is defined as , , , , , also under the condition, then is defined as , , . Under the homeomorphism map and . This proves that the diagram is commutative.

Also, the corresponding relations are described as that is,

Theorem 5. *Let be a fuzzy hyperboloid which is homeomorphic to , and a limit fuzzy retraction. Then, there is an induced limit fuzzy retraction such that the following diagram is commutative:
*(26)

*Proof. *Since and , under the homeomorphism map and . This proves that the diagram is commutative. Also, the corresponding relations are presented as

Theorem 6. *If the fuzzy deformation retract of the fuzzy hyperboloid is , the fuzzy retraction of is , and the limit of the fuzzy folding of is . Then there are induced fuzzy deformation retract, fuzzy retraction, and the limit of the fuzzy foldings such that the following diagram is commutative.*

*Proof. * Let the fuzzy deformation retract of be ; the fuzzy retraction of is defined by , , the fuzzy deformation retract of is , the fuzzy retraction of is given by , and , is a -dimensional space. Hence, the following diagram is commutative,
(28)
that is,.

Theorem 7. *Let be the fuzzy hyperboloid, then the relation between the fuzzy folding and the limit of the fuzzy retractions is discussed from the following commutative diagram.*

*Proof. *Let the fuzzy folding is , the limit of the fuzzy retractions of and are and , and . Then, the following diagram is commutative:
(29)
that is, , and the corresponding relations between the two chains of fuzzy folding and the limit of fuzzy retraction are given by

Theorem 8. *Let the fuzzy retraction of is , , and the fuzzy folding of is , then*(i)(ii)*. *

*Proof. *(i) Let the fuzzy retraction of the fuzzy hyperboloid in model be ; , the fuzzy retraction of is folding of is , and the fuzzy . Then ).

(ii) Let and are the compositions between the fuzzy retractions and the fuzzy foldings of into itself. Also, are the homeomorphisms. Then
(31)

Theorem 9. *Given the fuzzy deformation retract of model is ; the limit of the fuzzy folding of is . Then, the following diagram is commutative.*

* Proof. *Let the limit of the fuzzy folding of is , the fuzzy deformation retract of onto is , the limit of the fuzzy folding of is , and the fuzzy deformation retract of onto is . Hence
(32)
that is,

Theorem 10. *The composition of fuzzy deformation retracts of fuzzy hyperboloid model is a minimal retraction.*

* Proof. *Now consider the following fuzzy continuous map , such that = , then it is easy to see that
The fuzzy deformation retract of the fuzzy circle onto minimal fuzzy retraction is given in polar coordinates by
that is, is a fuzzy minimal retraction.

Theorem 11. *Let be a fuzzy hyperboloid in model which is homeomorphic to , : , the fuzzy retraction , and the limit fuzzy folding of is . Then there are induced fuzzy retraction, limit fuzzy folding, and homeomorphism map such that the following diagram is commutative:
*(35)

*Proof. *Let the homeomorphism map, and , also, , the fuzzy retraction of be ; the limit fuzzy folding of is given by , and . This proves that the diagram is commutative.

Theorem 12. * If the limit fuzzy folding of the fuzzy hyperboloid is , the fuzzy retraction of is and the homeomorphism map of is . Then there are induced limit fuzzy retractions, limit fuzzy folding, and homeomorphism map such that the following diagram is commutative:
*(36)

*Proof. *Consider the limit fuzzy folding of the fuzzy hyperboloid is , the fuzzy retraction of is , and the homeomorphism map of is , the limit fuzzy retraction of is , the limit fuzzy folding of is , and . This proves that the diagram is commutative.

Theorem 13. *Let and be a fuzzy retraction, and also a fuzzy folding. Then, there is induced limit fuzzy folding such that the following diagram is commutative:
*(37)

*Proof. *Let , and the fuzzy folding , also , and the limit fuzzy folding . This proves that the diagram is commutative.

Theorem 14. *If the fuzzy retraction of the fuzzy hyperboloid is , the fuzzy deformation retract of is , and the fuzzy deformation retract of is . Then there are induced end limit fuzzy retractions and fuzzy folding such that the following diagram is commutative.*

*Proof. *Let and also, ; the end limits fuzzy retractions of are ; the end limits fuzzy retractions of are , and , then the following diagram is commutative,
(38)
that is,

Also, the corresponding relation is described by the two induced chains, that is,

Theorem 15. *Let be the fuzzy hyperboloid, then the relations between the fuzzy deformation retract and the limit fuzzy folding is discussed from the following commutative diagram.*

*Proof. *Let the fuzzy deformation retract ; the limit fuzzy folding of and is and , and . Then, we have the following diagram,
(40)

that is, .

Also, the corresponding relations are described by the two induced chains, that is,

#### 3. Conclusion

In the present paper, we obtain and study new types of fuzzy retractions of model. Also, we deduced new types of fuzzy deformation retract of model. The relations between the fuzzy folding and the fuzzy deformation retracts of model is obtained. New types of minimal fuzzy retraction of model is also presented. New types of homotopy maps are described. The isometric and topological fuzzy folding in each case and the relation between the fuzzy deformation retract after and before fuzzy folding have been obtained. Types of conditional fuzzy folding of model are described.

#### References

- B. Schmeikal,
*Primordial Space—The Metric Case*, Nova Science, New York, NY, USA, 2010. - F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, and P. Zampetti,
*Geometry of Minkowski Space-Time*, Birkhauser, Boston, Mass, USA, 2011. - F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, and P. Zampetti,
*The Mathematics of Minkowski Space-Time*, Birkhauser, Boston, Mass, USA, 2008. - J. B. Hartle,
*Gravity, An Introduction to Einstein's General Relativity*, Addison-Wesley, New York, NY, USA, 2003. - J. B. Griffiths and J. Podolsky,
*Exact Space-Times in Einstein's General Relativity*, Cambridge University Press, Cambridge, UK, 2009. - N. Straumann,
*General Relativity with Application to Astrophysics*, Springer, New York, NY, USA, 2004. - A. E. El-Ahmady, “Folding of fuzzy hypertori and their retractions,”
*Proceedings of the Mathematical and Physical Society of Egypt*, vol. 85, no. 1, pp. 1–10, 2007. - A. E. El-Ahmady, “Limits of fuzzy retractions of fuzzy hyperspheres and their foldings,”
*Tamkang Journal of Mathematics*, vol. 37, no. 1, pp. 47–55, 2006. View at Scopus - G. L. Naber,
*Topology, Geometry and Gauge Fields: Foundations*, Texts in Applied Mathematics 25, Springer, New York, NY, USA, 2nd edition, 2011. - G. L. Naber,
*Topology, Geometry and Gauge Fields: Interactions, Applied Mathematics Sciences*, Springer, New York, NY, USA, 2nd edition, 2011. - M. Reid and B. Szendroi,
*Topology and Geometry*, Cambridge University Press, Cambridge, UK, 2005. - A. E. El-Ahmady, “The variation of the density functions on chaotic spheres in chaotic space-like Minkowski space time,”
*Chaos, Solitons and Fractals*, vol. 31, no. 5, pp. 1272–1278, 2007. View at Publisher · View at Google Scholar · View at Scopus - A. E. El-Ahmady and A. El-Araby, “On fuzzy spheres in fuzzy Minkowski space,”
*Nuovo Cimento della Societa Italiana di Fisica B*, vol. 125, no. 10, pp. 1153–1160, 2010. View at Publisher · View at Google Scholar · View at Scopus - A. E. El-Ahmady, “Fuzzy Lobachevskian space and its folding,”
*The Journal of Fuzzy Mathematics*, vol. 12, no. 2, pp. 609–614, 2004. - A. E. El-Ahmady, “Fuzzy folding of fuzzy horocycle,”
*Circolo Matematico Di Palermo Serie II*, vol. 53, no. 3, pp. 443–450, 2004. - A. P. Balachandran, S. Kurkcuoglu, and S. Vaidya,
*Lectures on Fuzzy and Fuzzy Susy Physics*, World Scientific Publishing, New York, NY, USA, 2007. - L. A. Zadeh,
*Fuzzy Sets and Their Application to Congnitive and Decision*, Academic Press, New York, NY, USA, 1975. - N. Palaniappan,
*Fuzzy Topology*, Alpha Science International, London, UK, 2005.