# Separability of Nonassociative Algebras with Metagroup Relations

## Abstract

**:**

## 1. Introduction

## 2. Separable Nonassociative Algebras

**Definition**

**1.**

**Ψ**be a (proper or improper) subgroup in the center $\mathcal{C}\left(G\right)$ of a metagroup G, let 1 denote a unit in $\mathcal{T}$, $\phantom{\rule{3.33333pt}{0ex}}e$ be a unit in G and let

**Proposition**

**1.**

**Proof.**

**Definition**

**2.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

## 3. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Metagroups

## References

- Bourbaki, N. Algèbre; Springer: Berlin, Germany, 2007; Chapter 1–3. [Google Scholar]
- Bourbaki, N. Algèbre homologique. In Algèbre; Springer: Berlin, Germany, 2007; Chapter 10. [Google Scholar]
- Florence, M. On higher trace forms of separable algebras. Arch. Math.
**2011**, 97, 247–249. [Google Scholar] [CrossRef] - Georgantas, G.T. Derivations in central separable algebras. Glasgow Math. J.
**1978**, 19, 75–77. [Google Scholar] [CrossRef] [Green Version] - Mazur, M.; Petrenko, B.V. Separable algebras over infinite fields are 2-generated and finitely presented. Arch. Math.
**2009**, 93, 521–529. [Google Scholar] [CrossRef] [Green Version] - Montgomery, S.; Smith, M.K. Algebras with a separable subalgebra whose centralizer satisfies a polynomial identity. Commun. Algebra
**1975**, 3, 151–168. [Google Scholar] [CrossRef] - Van Oystaeyen, F. Separable algebras. In Handbook of Algebra; Hazewinkel, M., Ed.; Elsevier: Amsterdam, The Netherlands, 2000; Volume 2, pp. 463–505. [Google Scholar]
- Pierce, R.S. Associative Algebras; Springer: New York, NY, USA, 1982. [Google Scholar]
- Rumynin, D.A. Cohomomorphisms of separable algebras. Algebra Log.
**1994**, 33, 233–237. [Google Scholar] [CrossRef] - Bredon, G.E. Sheaf Theory; McGraw-Hill: New York, NY, USA, 2012. [Google Scholar]
- Cartan, H.; Eilenberg, S. Homological Algebra; Princeton University Press: Princeton, NJ, USA, 1956. [Google Scholar]
- Hochschild, G. On the cohomology theory for associative algebras. Ann. Mathem.
**1946**, 47, 568–579. [Google Scholar] [CrossRef] - Pommaret, J.F. Systems of Partial Differential Equations and Lie Pseudogroups; Gordon and Breach Science Publishers: New York, NY, USA, 1978. [Google Scholar]
- Dickson, L.E. The Collected Mathematical Papers; Chelsea Publishing Co.: New York, NY, USA, 1975; Volume 1–5. [Google Scholar]
- Gürsey, F.; Tze, C.-H. On the Role of Division, Jordan and Related Algebras in Particle Physics; World Scientific Publication Co.: Singapore, 1996. [Google Scholar]
- Kantor, I.L.; Solodovnikov, A.S. Hypercomplex Numbers; Springer: Berlin, Germany, 1989. [Google Scholar]
- Krausshar, R.S. Generalized Analytic Automorphic Forms in Hypercomplex Spaces; Birkhäuser: Basel, Switzerlnad, 2004. [Google Scholar]
- Ludkowski, S.V. Integration of vector Sobolev type PDE over octonions. Complex Var. Elliptic Equat.
**2016**, 61, 1014–1035. [Google Scholar] - Ludkovsky, S.V. Normal families of functions and groups of pseudoconformal diffeomorphisms of quaternion and octonion variables. J. Math. Sci. N. Y.
**2008**, 150, 2224–2287. [Google Scholar] [CrossRef] - Ludkovsky, S.V.; Sprössig, W. Ordered representations of normal and super-differential operators in quaternion and octonion Hilbert spaces. Adv. Appl. Clifford Alg.
**2010**, 20, 321–342. [Google Scholar] - Ludkovsky, S.V.; Sprössig, W. Spectral theory of super-differential operators of quaternion and octonion variables. Adv. Appl. Clifford Alg.
**2011**, 21, 165–191. [Google Scholar] - Nichita, F.F. Unification theories: New results and examples. Axioms
**2019**, 8, 60. [Google Scholar] [CrossRef] [Green Version] - Schafer, R.D. An Introduction to Nonassociative Algebras; Academic Press: New York, NY, USA, 1966. [Google Scholar]
- Shang, Y. Lie algebraic discussion for affinity based information diffusion in social networks. Open Phys.
**2017**, 15, 705–711. [Google Scholar] [CrossRef] - Shang, Y. A Lie algebra approach to susceptible-infected-susceptible epidemics. Electr. J. Differ. Equat.
**2012**, 233, 1–7. [Google Scholar] - Ludkowski, S.V. Cohomology theory of nonassociative algebras with metagroup relations. Axioms
**2019**, 8, 78. [Google Scholar] [CrossRef] [Green Version] - Ludkowski, S.V. Automorphisms and derivations of nonassociative C
^{*}algebras. Linear Multilinear Algebra**2019**, 67, 1531–1538. [Google Scholar] [CrossRef] - Ludkowski, S.V. Smashed and twisted wreath products of metagroups. Axioms
**2019**, 8, 127. [Google Scholar] [CrossRef] [Green Version] - Jacobson, N. Structure and Representations of Jordan Algebras; Colloquium Publications; American Mathematical Society: Rhode Island, NY, USA, 1968; p. 39. [Google Scholar]
- Zaikin, B.A.; Bogadarov, A.Y.; Kotov, A.F.; Poponov, P.V. Evaluation of coordinates of air target in a two-position range measurement radar. Russ. Technol. J.
**2016**, 4, 65–72. [Google Scholar] - Blahut, R.E. Algebraic Codes for Data Transmission; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Magomedov, S.G. Assessment of the impact of confounding factors in the performance information security. Russ. Technol. J.
**2017**, 5, 47–56. [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] [Green Version] - 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]

© 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.
Separability of Nonassociative Algebras with Metagroup Relations. *Axioms* **2019**, *8*, 139.
https://doi.org/10.3390/axioms8040139

**AMA Style**

Ludkowski SV.
Separability of Nonassociative Algebras with Metagroup Relations. *Axioms*. 2019; 8(4):139.
https://doi.org/10.3390/axioms8040139

**Chicago/Turabian Style**

Ludkowski, Sergey V.
2019. "Separability of Nonassociative Algebras with Metagroup Relations" *Axioms* 8, no. 4: 139.
https://doi.org/10.3390/axioms8040139