Yang-Baxter Equation in Lattice Effect Algebras

Abstract

In this paper, we present lattice effect algebra by giving certain definitions and conceptions as to lattice effect algebra. We present new solutions to the set-theoratical Yang-Baxter equation by using lattice effect algebras.

Keywords

• Yang–Baxter Equation,Lattice Effect Algebras

References

1. [1] Baxter, R.J., Exactly Solved Models in Statistical Mechanics, Academy Press: London, UK, 1982.
2. [2] Baxter, R.J., Partition function of the eight-vertex lattice model. Annals of Physics 1972, (70)1, 193–228.
3. [3] Borzooei, R.A., Dvurecenskij, A., and Sharafi, A.H. Material Implications in Lattice Effect Algebras, Information Sciences 433-434 2018 , 233-240.
4. [4] Botur, M., Chajda, I., Halas, R., K¨uhr, J., Paseka, J., “Algebraic Methods in Quantum Logic”, 1st ed. Palack´y University, Olomouc, 2014.
5. [5] Dalla Chiara, M. L, Giuntini, R., Greechie, R., “Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics” Springer, Netherlands, 2004.
6. [6] Drinfeld, V. G., On unsolved problems in quantum group theory. Quantum Groups, Lecture Notes Math. 1510, Springer-Verlag, Berlin, 1992, 1–8.
7. [7] Dvurecenskij, A. , Pulmannova, S., New Trends in Quantum Structures, Springer, Netherlands, 2000.
8. [8] Foulis, D.J., Bennett, M.K., Effect Algebras and Unsharp Quantum Logics, Found. Phys. 24 (1994), 1331–1352.

9. [9] Kassel, C., Quantum Groups, Graduate Text in Mathematics 155, Springer-Verlag, New York, 1995.
10. [10] Nichita, F.F., Special Issue “Hopf Algebras, Quantum Groups and Yang–Baxter Equations. Available online: http://www.mdpi.com/journal/axioms/special issues/hopf algebras 2014.
11. [11] Nichita, F.F., Introduction to the Yang-Baxter Equation with Open Problems Axioms 2012, 4, 423–435.
12. [12] Oner, T. Kalkan, T., Yang-Baxter Equations in MTL-algebras.. Bulletin of the International Mathematical Virtual Institute 2019, 9, 599–607.
13. [13] Oner, T. Katıcan, T., On solution to the Set-Theoretical Yang-Baxter Equation via BL-Algebras. Bulletin of the International Mathematical Virtual Institute 2019, 9, 207–217.
14. [14] Oner, T., Katıcan, T., On solutions to the Set-Theoretical Yang-Baxter Equation in Wasjberg-Algebras. Axioms 2018, 7, 6 .
15. [15] Perk, J.H.H., Au, Y.H., Yang-Baxter Equations. In Encyclopedia of Mathematical Physics; Franc¸oise, J.-P., Naber, G.L., Tsou S.T., Eds.; Elseiver: Oxford, UK 2006, 5, 465–473.
16. [16] Yang, C.N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Physical Review Letters 1967, 19, 1312–1315.