# Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings

## Abstract

**:**

## 1. Introduction

#### 1.1. Compactification and Projections

#### 1.2. Decomposition and Soft Sets

#### 1.3. Motivation

## 2. Preliminary Concepts

## 3. Decomposition Varieties

#### 3.1. Non-Projective Decomposition: Definitions

#### 3.1.1. Topological Space Partition

#### 3.1.2. Group Homeomorphisms

#### 3.1.3. $G-$Partition

#### 3.1.4. $G-$Decomposition

#### 3.1.5. ${\mathsf{\Pi}}_{G}-$Fiber

#### 3.1.6. $\ast -$Locality of ${\mathsf{\Pi}}_{G}$

#### 3.1.7. ${\mathsf{\Pi}}_{G}-$Homeomorphism

#### 3.1.8. Homeomorphic $\ast -$Locality

#### 3.1.9. Schoenflies Homeomorphic Embeddings

#### 3.1.10. Decomposed Group Embedding

#### 3.2. Topological Decomposition with Projection: Definitions

#### 3.2.1. Gravity of Decomposition

#### 3.2.2. Monotone Class of Decomposition

#### 3.2.3. Projection of Decomposition

#### 3.2.4. Noncommutative Projection

#### 3.2.5. Finiteness of Translated Projection

#### 3.2.6. Inflationary Bounded Translation

## 4. Analytical Properties of Decomposition Varieties

#### 4.1. Properties of Non-Projective Variety

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

#### 4.2. Properties of Projective Variety of Decomposition

**Theorem**

**13.**

**Proof.**

**Theorem**

**14.**

**Proof.**

**Theorem**

**15.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**16.**

**Proof.**

**Theorem**

**17.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**18.**

**Proof.**

**Theorem**

**19.**

**Proof.**

**Theorem**

**20.**

**Proof.**

**Remark**

**2.**

**Theorem**

**21.**

**Proof.**

**Theorem**

**22.**

**Proof.**

**Lemma**

**3.**

**Proof.**

## 5. Application Aspects and Comparison

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Vasco, M.I.G.; Steinwandt, R. Group Theoretic Cryptography; Chapman & Hall/CRC: Hoboken, NJ, USA, 2015; ISBN 9781584888369. [Google Scholar]
- Chen, Y.; Feng, J. Group-Theoretic Exploitations of Symmetry in Novel Prestressed Structures. Symmetry
**2018**, 10, 229. [Google Scholar] [CrossRef] [Green Version] - Dutka, J.; Ivanov, A. Topologizable structures and Zariski topology. Algebra Univers.
**2018**, 79, 72. [Google Scholar] [CrossRef] - Xiao, Z.; Sanchez, I.; Tkachenko, M. Topological groups whose closed subgroups are separable, and the product operation. Topol. Its Appl.
**2019**, 259, 365–377. [Google Scholar] [CrossRef] - Cheltsov, I. Points in projective spaces and applications. J. Differ. Geom.
**2009**, 81, 575–599. [Google Scholar] [CrossRef] - Hirschfeld, J.W.P.; Thas, J.A. Open problems in finite projective spaces. Finite Fields Their Appl.
**2015**, 32, 44–81. [Google Scholar] [CrossRef] - Uspenskij, V. Compactifications of Topological Groups. In Proceedings of the Ninth Prague Topological Symposium, Prague, Czech Republic, 19–25 August 2001; pp. 331–346. [Google Scholar]
- Chu, H. Compactification and Duality of Topological Groups. Trans. Am. Math. Soc.
**1966**, 123, 310–324. [Google Scholar] [CrossRef] - Pontrjagin, L. The Theory of Topological Commutative Groups. Ann. Math.
**1934**, 35, 361–388. [Google Scholar] [CrossRef] - Lau, A.T.; Milnes, P.; Pym, J.S. Compactifications of locally compact groups and closed subgroups. Trans. Am. Math. Soc.
**1992**, 329, 97–115. [Google Scholar] [CrossRef] - Giannelli, E.; Wildon, M. Foulkes modules and decomposition numbers of the symmetric group. J. Pure Appl. Algebra
**2015**, 219, 255–276. [Google Scholar] [CrossRef] [Green Version] - Gleason, A.M. Projective topological spaces. Ill. J. Math.
**1958**, 2, 482–489. [Google Scholar] [CrossRef] - Insel, A.J. A Decomposition Theorem for Topological Group Extensions. Pac. J. Math.
**1971**, 36, 357–378. [Google Scholar] [CrossRef] [Green Version] - Weichsel, P.M. A Decomposition Theory for Finite Groups with Applications to p-Groups. Trans. Am. Math. Soc.
**1962**, 102, 218–226. [Google Scholar] - Romanovskii, N.S. Decomposition of a Group over an Abelian Normal Subgroup. Algebra Log.
**2016**, 55, 315–326. [Google Scholar] [CrossRef] - Munthe-Kaas, H.Z.; Quispel, G.R.W.; Zanna, A. Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms. Found. Comput. Math.
**2001**, 1, 297–324. [Google Scholar] [CrossRef] - Sawyer, P. Computing the Iwasawa Decomposition of the Classical Lie Groups of Noncompact type using the QR Decomposition. Linear Algebra Its Appl.
**2016**, 493, 573–579. [Google Scholar] [CrossRef] [Green Version] - Benzi, M.; Razouk, N. On the Iwasawa Decomposition of a Symplectic Matrix. Appl. Math. Lett.
**2007**, 20, 260–265. [Google Scholar] [CrossRef] [Green Version] - Huang, H.; Tam, T.Y. An Asymptotic Result on the a-component in the Iwasawa Decomposition. J. Lie Theory
**2007**, 17, 469–479. [Google Scholar] - Kostant, B. On Convexity, the Weyl Group and the Iwasawa Decomposition. Annales Scientifiques de L’ecole Normale Superieure
**1973**, 6, 413–455. [Google Scholar] [CrossRef] [Green Version] - Georgiou, D.N.; Megaritis, A.C.; Petropoulos, V.I. On Soft Topological Spaces. Appl. Math. Inf. Sci.
**2013**, 7, 1889–1901. [Google Scholar] [CrossRef] - Sayed, O.R.; Hassan, N.; Khalil, A.M. A Decomposition of Soft Continuity in Soft Topological Spaces. Afr. Math.
**2017**, 28, 887–898. [Google Scholar] [CrossRef] - Kandil, A.; Tantawy, O.A.E.; El-Sheikh, S.A.; El-Latif, A.M.A. γ-operation and Decompositions of some forms of Soft Continuity in Soft Topological Spaces. Ann. Fuzzy Math. Inform.
**2014**, 7, 181–196. [Google Scholar] - Acikgoz, A.; Noiri, T.; Yuksel, S. A Decomposition of Continuity in Ideal Topological Spaces. Acta Math. Hung.
**2004**, 105, 285–289. [Google Scholar] [CrossRef] - Cristea, I. Reducibility in hypergroups—The crisp and fuzzy cases. In Proceedings of the 12th AHA Congress at Democritus University of Thrace, Xanthi, Greece, 2–7 September 2014. [Google Scholar] [CrossRef]
- Chen, B.; Du, C.Y.; Wang, R. The groupoid structure of groupoid morphisms. J. Geom. Phys.
**2019**, 45, 103486. [Google Scholar] [CrossRef] [Green Version] - Weiss, W. Partitioning topological spaces. Math. Ramsey Theory
**1990**, 5, 154–171. [Google Scholar] - Bourbaki, N. A Panorama of Pure Mathematics; Academic Press: Cambridge, MA, USA, 1982; ISBN 0-12-215560-2. [Google Scholar]
- Guggenheimer, H. The Jordan and Schoenflies Theorems in Axiomatic Geometry. Am. Math. Mon.
**1978**, 85, 753–756. [Google Scholar] [CrossRef] - Inui, T.; Tanabe, Y.; Onodera, Y. Group Theory and Its Applications in Physics; Springer Series in Solid-State Sciences 78; Springer: Berlin/Heidelberg, Germany, 1980; ISBN 978-3-540-60445-7. [Google Scholar]

Decomposition Varieties | Partitioning Structure | Normality Property of Subgroup | Identity Placement | Projection/Fiber |
---|---|---|---|---|

Semidirect product | ${H}_{1}\otimes {H}_{2}$ | ${H}_{1}$ is normal | ${H}_{1}\cap {H}_{2}=\left\{e\right\}$ | Projection |

Direct product | ${H}_{1}\times {H}_{2}$ | ${H}_{1},{H}_{2}$ both normal | ${e}_{H1}\ne {e}_{H2}$ | Projection |

Topological decomposition | $\{A,B,X\backslash (\overline{A\cup B})\}$ | Not applicable | $A\cap B=\varphi ,e\in X$ | Topological fiber and projection |

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Bagchi, S.
Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings. *Symmetry* **2020**, *12*, 450.
https://doi.org/10.3390/sym12030450

**AMA Style**

Bagchi S.
Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings. *Symmetry*. 2020; 12(3):450.
https://doi.org/10.3390/sym12030450

**Chicago/Turabian Style**

Bagchi, Susmit.
2020. "Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings" *Symmetry* 12, no. 3: 450.
https://doi.org/10.3390/sym12030450