# Mathematics and Poetry · Yang–Baxter Equations, Boolean Algebras, and BCK-Algebras

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## Abstract

**:**

## 1. Introduction

## 2. Rudiments of BCK-Algebras

**Definition**

**1.**

- (i)
- $(x\to y)\to \left(\right(y\to z)\to (x\to z\left)\right)=1$,
- (ii)
- $x\to \left(\right(x\to y)\to y)=1$,
- (iii)
- $x\to x=1$,
- (iv)
- $x\to 1=1$,
- (v)
- $x\to y=1\text{}\mathit{and}\text{}y\to x=1\hspace{1em}imply\hspace{1em}x=y$.

**Lemma**

**1.**

**Definition**

**2.**

**Lemma**

**2.**

- (a)
- $x\le y\Rightarrow y\to z\le x\to z$,
- (b)
- $x\le y\Rightarrow z\to x\le z\to y$,
- (c)
- $x\to (y\to z)=y\to (x\to z)$,
- (d)
- $y\le x\to y$,
- (e)
- $1\to x=x$,
- (f)
- $x\to y\le (z\to x)\to (z\to y)$,
- (g)
- $\left(\right(x\to y)\to y)\to y=x\to y$,

**Definition**

**3.**

**Lemma**

**3.**

**Proof.**

**Definition**

**4.**

**Definition**

**5.**

**Theorem**

**1.**

- (i)
- $a\bigsqcup a=a,$
- (ii)
- $a\bigsqcup b=b$ and $b\bigsqcup a=a$ imply $a=b$,
- (iii)
- $a\bigsqcup b=(a\bigsqcup b)\bigsqcup b=a\bigsqcup (a\bigsqcup b)=b\bigsqcup (a\bigsqcup b),$
- (iv)
- $(a\bigsqcup c)\bigsqcup \left(\right(a\bigsqcup b)\bigsqcup c)=(a\bigsqcup b)\bigsqcup c,$
- (v)
- $a\bigsqcup 1=1,$
- (vi)
- ${N}_{a}\left(a\right)=1,\hspace{1em}{N}_{a}\left(1\right)=a,$
- (vii)
- $a\bigsqcup b={N}_{b}\left({N}_{b}(a\bigsqcup b)\right)={N}_{b}({N}_{b}(a\bigsqcup b)\bigsqcup b),$
- (viii)
- ${N}_{b}(a\bigsqcup b)\bigsqcup {N}_{b\bigsqcup c}((a\bigsqcup c)\bigsqcup (b\bigsqcup c))={N}_{b\bigsqcup c}((a\bigsqcup c)\bigsqcup (b\bigsqcup c)),$
- (ix)
- ${N}_{b\bigsqcup c}({N}_{c}(a\bigsqcup c)\bigsqcup (b\bigsqcup c))={N}_{a\bigsqcup c}({N}_{c}(b\bigsqcup c)\bigsqcup (a\bigsqcup c)),$
- (x)
- ${N}_{a}((a\bigsqcup b)\bigsqcup a)={N}_{a}(a\bigsqcup b).$

**Lemma**

**4.**

**Lemma**

**5.**

- (i)
- $\mathcal{A}$ is commutative;
- (ii)
- $(A;\bigsqcup )$ is a directoid;
- (iii)
- $(A;\bigsqcup )$ is a a join-semilattice.

**Theorem**

**2.**

**Definition**

**6.**

- If it verifies$$x\to (y\to z)=(x\to y)\to (x\to z)$$
- If it verifies$$(x\to y)\to z=(x\to z)\to (y\to z)$$

**Definition**

**7.**

- (i)
- Δ is coassociative (i.e., $(\Delta \times I)\circ \Delta =(I\times \Delta )\circ \Delta $),
- (ii)
- $\text{}{a}_{1}\vee {a}_{2}=a\hspace{1em}\forall a\in C$,
- (iii)
- $\text{}\Delta \left(0\right)=\text{}(0,\text{}0)$ and
- (iv)
- $\Delta (a\vee b)=\text{}\Delta \left(a\right)\vee \Delta \left(b\right)\hspace{1em}\forall a,b\in C\text{}$ (Notice that this equality takes place in $C\times C$).

## 3. Perspectives on the Yang–Baxter Equation in BCK-Algebras

**Definition**

**8**

**Lemma**

**7**

**Definition**

**9**

**Proposition**

**1**

- (1)
- $(0\to 0)\to x=x$
- (2)
- $(x\to 0)\to 0=x$
- (3)
- $(z\to 0)\to y=(y\to 0)\to z$
- (4)
- $x\to (y\to x)=1$.

**Lemma**

**8.**

**Lemma**

**9.**

**Lemma**

**10.**

**Proof.**

**Lemma**

**11.**

**Proof.**

**Lemma**

**12.**

**Proof.**

**Example**

**1.**

→ | 0 | x | 1 |

0 | 1 | 1 | 1 |

x | x | 1 | 1 |

1 | 0 | x | 1 |

**Proof.**

**Example**

**2.**

**Lemma**

**13.**

**Proof.**

**Lemma**

**14.**

**Proof.**

**Lemma**

**15.**

**Proof.**

**Lemma**

**16.**

**Proof.**

**Lemma**

**17.**

**Proof.**

**Lemma**

**18.**

**Proof.**

**Lemma**

**19.**

**Proof.**

**Theorem 3.**

**Remark**

**1.**

**Theorem 4.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 4. Poetry and the Yang–Baxter Equation

- like the Kiang river,
- like the Han river,
- massive like the mountains,
- voluminously flowing like the rivers.”

- how can we for a single day
- do without these men”

- “A piece of literature
- Is meant for the millennium
- But its ups and downs are known
- Already in the author’s heart.”

- (…)
- The small particle
- was captured…
- The common piece of information…
- The two streams
- arrived on my table from overseas
- were unified…

- A POST-MODERN MANIFEST

- Once… after a promenade,
- I gorged a “pomegranate”:

- Abstract cocktail of notes,
- mixed with books,
- flowers and clothes,
- resting on my desks,
- falling from the shelves,
- rolling on the chair,
- flying in the air…

- From the open volumes,
- on inevitable social inequalities,
- to the open problems,
- on classical means inequalities…

- Subtle metaphors,
- musical measures,
- philosophical concepts,
- mathematical models,
- entwined structures,
- historical phrases…

- Amalgamated groups…

- Kaleidoscopic traces…

## 5. Poetry and Mathematics—Other Aspects

- CHANGE OF GUARDS IN THE WINTER

- The silver lake,
- Cooked as a steak…
- Icicles serving,
- Drinking and dancing…

- “Look at those marvels:
- Windows with flowers…”
- “Take with a fork
- Some fine art work!”

- Wine from the steam
- Snows on the realm…
- The wild flame bites
- The icy coulds…

- The silver lake, 6 (first team)
- Cooked as a steak… −1 (second team)
- Icicles serving, 5 (first team)
- Drinking and dancing… −2 (second team)
- “Look at those marvels: 4 (first team)
- Windows with flowers…” −3 (second team)
- “Take with a fork 3 (first team)
- Some fine art work!” −4 (second team)
- Wine from the steam 2 (first team)
- Snows on the realm… −5 (second team)
- The wild flame bites 1 (first team)
- The icy coulds… −6 (second team)

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Yang, C.N. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett.
**1967**, 19, 1312–1315. [Google Scholar] [CrossRef] - Baxter, R.J. Exactly Solved Models in Statistical Mechanics; Academy Press: London, UK, 1982. [Google Scholar]
- Baxter, R.J. Partition function of the eight-vertex lattice model. Ann. Phys.
**1972**, 70, 193–228. [Google Scholar] [CrossRef] - Perk, J.H.H.; Au, Y.H. Yang-Baxter Equations. In Encyclopedia of Mathematical Physics; Françoise, J.-P., Naber, G.L., Tsou, S.T., Eds.; Elseiver: Oxford, UK, 2006; Volume 5, pp. 465–473. [Google Scholar]
- Nichita, F.F. Introduction to the Yang-Baxter Equation with Open Problems. Axioms
**2012**, 1, 33–37. [Google Scholar] [CrossRef] [Green Version] - Nichita, F.F. Unification Theories: New Results and Examples. Axioms
**2019**, 8, 60. [Google Scholar] [CrossRef] [Green Version] - Nichita, F.F. On Jordan (Co)algebras. Rev. Roum. Math. Pures Appl.
**2014**, 59, 401–409. [Google Scholar] - Massuyeau, G.; Nichita, F.F. Yang-Baxter operators arising from algebra structures and the Alexander polynomial of knots. Comm. Algebra
**2005**, 33, 2375–2385. [Google Scholar] [CrossRef] [Green Version] - Nichita, F.F. On Jordan Algebras and Unification Theories. Rev. Roum. Math. Pures Appl.
**2016**, 61, 305–316. [Google Scholar] - Oner, T.; Senturk, I.; Oner, G. An Independent Set of Axioms of MV-algebras and Solutions of the Set-Theoretical Yang-Baxter Equation. Axioms
**2017**, 6, 17. [Google Scholar] [CrossRef] [Green Version] - Oner, T.; Kalkan, T. Yang-Baxter Equations in MTL-Algebras. Bull. Int. Math. Virtual Inst.
**2020**, 10, 599–607. [Google Scholar] - Oner, T.; Kalkan, T.; Gursoy, N. Weak Implication Algebra and Solutions to the Set-Theoretical Yang-Baxter Equation. J. Int. Math. Virtual Inst.
**2020**, 10, 139–156. [Google Scholar] - Oner, T.; Kalkan, T.; Ulker, A. Yang-Baxter Equation in Lattice Effect Algebras. Konuralp J. Math.
**2020**, 8, 106–113. [Google Scholar] - Iseki, K. An algebras related with a propositional calculus. Math. Jpn.
**1966**, 42, 26–29. [Google Scholar] [CrossRef] - Iseki, K.; Tanaka, S. An introduction to theory of BCK-algebras. Math. Jpn.
**1978**, 23, 1–26. [Google Scholar] - Chajda, I.; Kühr, J. Algebraic Structures Derived From BCK-algebras. Miskolc Math. Notes
**2007**, 8, 11–21. [Google Scholar] [CrossRef] - Nichita, F.F. Yang-Baxter Equations, Computational Methods and Applications. Axioms
**2015**, 4, 423–435. [Google Scholar] [CrossRef] [Green Version] - Nichita, F.F. On the Johnson–Tzitzeica Theorem, Graph Theory, and Yang-Baxter Equations. Symmetry
**2021**, 13, 2070. [Google Scholar] [CrossRef] - Solomon, M.; Nichita, F.F. On Transcendental Numbers: New Results and a Little History. Axioms
**2018**, 7, 15. [Google Scholar] - Yang, C.N. Banquet Speech. Available online: https://www.nobelprize.org/prizes/physics/1957/yang/speech/ (accessed on 10 March 2020).
- Nichita, F.F. Mathematics and Poetry · Unification, Unity, Union. Sci
**2020**, 2, 72. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems. Sci
**2021**, 3, 39. [Google Scholar] [CrossRef] - Calin, O. Statistics and Machine Learning Experiments in English and Romanian Poetry. Sci
**2020**, 2, 92. [Google Scholar] [CrossRef] - Crease, R.P.; Goldhaber, A.S. The Quantum Moment: How Planck, Bohr, Einstein, and Heisenberg Taught Us to Love Uncertainty; Hardcover; W. W. Norton Company: New York, NY, USA, 2014; 352p, ISBN1 9780393067927. ISBN2 10: 0393067920. [Google Scholar]
- O’Keefe, M. The Quantum Poet. Available online: https://www.symmetrymagazine.org/article/the-quantum-poet (accessed on 10 March 2020).

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**MDPI and ACS Style**

Kalkan, T.; Nichita, F.F.; Oner, T.; Senturk, I.; Terziler, M.
Mathematics and Poetry · Yang–Baxter Equations, Boolean Algebras, and BCK-Algebras. *Sci* **2022**, *4*, 16.
https://doi.org/10.3390/sci4020016

**AMA Style**

Kalkan T, Nichita FF, Oner T, Senturk I, Terziler M.
Mathematics and Poetry · Yang–Baxter Equations, Boolean Algebras, and BCK-Algebras. *Sci*. 2022; 4(2):16.
https://doi.org/10.3390/sci4020016

**Chicago/Turabian Style**

Kalkan, Tugce, Florin F. Nichita, Tahsin Oner, Ibrahim Senturk, and Mehmet Terziler.
2022. "Mathematics and Poetry · Yang–Baxter Equations, Boolean Algebras, and BCK-Algebras" *Sci* 4, no. 2: 16.
https://doi.org/10.3390/sci4020016