# Unification Theories: New Results and Examples

## Abstract

**:**

## 1. Introduction

## 2. Weak Ujla Structures, Dual Structures, Unification

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 3. Unification of (Co)Derivations and Applications

**Definition**

**4.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Definition**

**5.**

**Remark**

**1.**

**Theorem**

**6.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 4. Modified Yang–Baxter Equation

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

**Remark**

**8.**

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Nichita, F.F. Unification theories: Examples and Applications. Axioms
**2018**, 7, 85. [Google Scholar] [CrossRef] - Marcus, S.; Nichita, F.F. On Transcendental Numbers: New Results and a Little History. Axioms
**2018**, 7, 15. [Google Scholar] [CrossRef] - Smoktunowicz, A.; Smoktunowicz, A. Set-theoretic solutions of the Yang-Baxter equation and new classes of R-matrices. Linear Algebra Its Appl.
**2018**, 546, 86–114. [Google Scholar] [CrossRef] - Motegi, K.; Sakai, K. Quantum integrable combinatorics of Schur polynomials. arXiv
**2015**, arXiv:1507.06740. [Google Scholar] - Nichita, F.F. On Jordan algebras and unification theories. Rev. Roum. Math. Pures Appl.
**2016**, 61, 305–316. [Google Scholar] - Ardizzoni, A.; Kaoutit, L.E.; Saracco, P. Functorial Constructions for Non-associative Algebras with Applications to Quasi-bialgebras. arXiv
**2015**, arXiv:1507.02402. [Google Scholar] [CrossRef] - Nichita, F.F. Self-inverse Yang-Baxter operators from (co)algebra structures. J. Algebra
**1999**, 218, 738–759. [Google Scholar] [CrossRef] - Dascalescu, S.; Nichita, F.F. Yang-Baxter Operators Arising from (Co)Algebra Structures. Commun. Algebra
**1999**, 27, 5833–5845. [Google Scholar] [CrossRef] - Grinberg, D. Collected Trivialities On Algebra Derivations. Available online: http://www.cip.ifi.lmu.de (accessed on 16 May 2019).
- Bordemann, M. Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups. Commun. Math. Phys.
**1990**, 135, 201–216. [Google Scholar] [CrossRef] - Fulton, W. Eigenvalues, Invariant Factors, Highest Weights, and Schubert Calculus. Bull. New Ser. AMS
**2000**, 37, 209–249. [Google Scholar] [CrossRef] - Chiappinelli, R. What Do You Mean by “Nonlinear Eigenvalue Problems”? Axioms
**2018**, 7, 39. [Google Scholar] [CrossRef] - Ding, J.; Tian, H.Y. Solving the Yang–Baxter–like matrix equation for a class of elementary matrices. Comput. Math. Appl.
**2016**, 72, 1541–1548. [Google Scholar] [CrossRef] - Zhou, D.; Chen, G.; Ding, J. On the Yang-Baxter matrix equation for rank-two matrices. Open Math.
**2017**, 15, 340–353. [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**

Nichita, F.F. Unification Theories: New Results and Examples. *Axioms* **2019**, *8*, 60.
https://doi.org/10.3390/axioms8020060

**AMA Style**

Nichita FF. Unification Theories: New Results and Examples. *Axioms*. 2019; 8(2):60.
https://doi.org/10.3390/axioms8020060

**Chicago/Turabian Style**

Nichita, Florin F. 2019. "Unification Theories: New Results and Examples" *Axioms* 8, no. 2: 60.
https://doi.org/10.3390/axioms8020060