# Smashed and Twisted Wreath Products of Metagroups

## Abstract

**:**

## 1. Introduction

## 2. Nonassociative Metagroups

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

## 3. Smashed Products and Smashed Twisted Products of Metagroups

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Definition**

**2.**

**Remark**

**2.**

## 4. Smashed Twisted Wreath Products of Metagroups

**Lemma**

**5.**

**Proof.**

**Definition**

**3.**

**Corollary**

**1.**

**Remark**

**3.**

**Remark**

**4.**

**Lemma**

**6.**

**Proof.**

**Definition**

**4.**

**Theorem**

**5.**

**Proof.**

**Definition**

**5.**

**Theorem**

**6.**

**Proof.**

**Remark**

**5.**

**Definition**

**6.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

- Bruck, R.H. A Survey of Binary Systems; Springer: Berlin, Germany, 1971. [Google Scholar]
- Kakkar, V. Boolean loops with compact left inner mapping groups are profinite. Topol. Appl.
**2018**, 244, 51–54. [Google Scholar] [CrossRef] - Razmyslov, Y.P. Identities of Algebras and their Representations; Series Modern Algebra; Nauka: Moscow, Soviet Union, 1989; Volume 14. [Google Scholar]
- Vojtěchovský, P. Bol loops and Bruck loops of order pq up to isotopism. Finite Fields Appl.
**2018**, 52, 1–9. [Google Scholar] - Pickert, G. Projektive Ebenen; Springer: Berlin, Germany, 1955. [Google Scholar]
- Pickert, G. Doppelebenen und loops. J. Geom.
**1991**, 41, 133–144. [Google Scholar] [CrossRef] - Baez, J.C. The octonions. Bull. Am. Math. Soc.
**2002**, 39, 145–205. [Google Scholar] [CrossRef] - Bogolubov, N.N.; Logunov, A.A.; Oksak, A.I.; Todorov, I.T. General Principles of Quantum Field Theory; Nauka: Moscow, Russia, 1987. [Google Scholar]
- Bourbaki, N. Algebra; Springer: Berlin, Germany, 1989. [Google Scholar]
- Castro-Alvaredo, O.A.; Doyon, B.; Fioravanti, D. Conical twist fields and null polygonal Wilson loops. Nuclear Phys.
**2018**, B931, 146–178. [Google Scholar] [CrossRef] - Dickson, L.E. The Collected Mathematical Papers; Chelsea Publishing Co.: New York, NY, USA, 1975; Volumes 1–5. [Google Scholar]
- Kantor, I.L.; Solodovnikov, A.S. Hypercomplex Numbers; Springer: Berlin, Germany, 1989. [Google Scholar]
- Schafer, R.D. An Introduction to Nonassociative Algebras; Academic Press: New York, NY, USA, 1966. [Google Scholar]
- Frenod, E.; Ludkowski, S.V. Integral operator approach over octonions to solution of nonlinear PDE. Far East J. Math. Sci. (FJMS)
**2018**, 103, 831–876. [Google Scholar] [CrossRef] - Gürlebeck, K.; Sprössig, W. Quaternionic and Clifford Calculus for Physicists and Engineers; John Wiley and Sons: Chichester, UK, 1997. [Google Scholar]
- Gürsey, F.; Tze, C.-H. On the Role of Division, Jordan and Related Algebras in Particle Physics; World Scientific Publishing Co.: Singapore, 1996. [Google Scholar]
- Ludkowski, S.V. Integration of vector Sobolev type PDE over octonions. Complex Var. Elliptic Equ.
**2016**, 61, 1014–1035. [Google Scholar] - Ludkovsky, S.V.; Sprössig, W. Spectral theory of super-differential operators of quaternion and octonion variables. Adv. Appl. Clifford Algebras
**2011**, 21, 165–191. [Google Scholar] - Ludkovsky, S.V. Integration of vector hydrodynamical partial differential equations over octonions. Complex Var. Elliptic Equ.
**2013**, 58, 579–609. [Google Scholar] [CrossRef] - Ludkowski, S.V. Automorphisms and derivations of nonassociative C
^{*}algebras. Linear Multilinear Algebra**2019**, 67, 1531–1538. [Google Scholar] [CrossRef] - Ludkowski, S.V. Cohomology theory of nonassociative algebras. Axioms
**2019**, 8, 78. [Google Scholar] [CrossRef] - Betten, A. Twisted tensor product codes. Des. Codes Cryptogr.
**2008**, 47, 191–219. [Google Scholar] [CrossRef] - Fernández-López, M.; García-Río, E.; Kupeli, D.N.; Ünal, B. A curvature condition for a twisted product to be a warped product. Manuscripta Math.
**2001**, 106, 213–217. [Google Scholar] [CrossRef] - Mikaelian, V.H. The criterion of Shmel’kin and varieties generated by wreath products of finite groups. Algebra Logic
**2017**, 56, 108–115. [Google Scholar] [CrossRef] - Rudkovski, M.A. Twisted products of Lie groups. Sib. Math. J.
**1997**, 38, 969–977. [Google Scholar] [CrossRef] - Othman, M.I.A.; Marin, M. Effect of thermal loading due to laser pulse on thermoelastic porous medium under G-N theory. Results Phys.
**2017**, 7, 3863–3872. [Google Scholar] [CrossRef] - Kunen, K. Set Theory; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1980. [Google Scholar]
- Blahut, R.E. Algebraic Codes for Data Transmission; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Shum, K.P.; Ren, X.; Wang, Y. Semigroups on semilattice and the constructions of generalized cryptogroups. Southeast Asian Bull. Math.
**2014**, 38, 719–730. [Google Scholar] - Sigov, A.S.; Andrianova, E.G.; Zhukov, D.O.; Zykov, S.V.; Tarasov, I.E. Quantum informatics: Overview of the main achievements. Russ. Technol. J.
**2019**, 7, 5–37. [Google Scholar] [CrossRef] - Gilbert, J.E.; Murray, M.A.M. Clifford Algebras and Dirac Operators in Harmonic Analysis; Cambridge Studies in Advanced Mathematics Book 26; Cambridge University Press: Cambridge, UK, 1991. [Google Scholar]
- Girard, P.R. Quaternions, Clifford Algebras and Relativistic Physics; Birkhäuser: Basel, Switzerland, 2007. [Google Scholar]
- Ludkowski, S.V. Manifolds over Cayley-Dickson algebras and their immersions. Rendic. dell’Istit. di Matem. dell’Univer. di Trieste. Nuova Ser.
**2013**, 45, 11–22. [Google Scholar] - Ludkovsky, S.V. Normal families of functions and groups of pseudoconformal diffeomorphisms of quaternion and octonion variables. J. Math. Sci.
**2008**, 150, 2224–2287. [Google Scholar] [CrossRef] - Ludkovsky, S.V. Functions of several Cayley-Dickson variables and manifolds over them. J. Math. Sci.
**2007**, 141, 1299–1330. [Google Scholar] [CrossRef]

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ludkowski, S.V.
Smashed and Twisted Wreath Products of Metagroups. *Axioms* **2019**, *8*, 127.
https://doi.org/10.3390/axioms8040127

**AMA Style**

Ludkowski SV.
Smashed and Twisted Wreath Products of Metagroups. *Axioms*. 2019; 8(4):127.
https://doi.org/10.3390/axioms8040127

**Chicago/Turabian Style**

Ludkowski, Sergey V.
2019. "Smashed and Twisted Wreath Products of Metagroups" *Axioms* 8, no. 4: 127.
https://doi.org/10.3390/axioms8040127