A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System

Abstract

In the present manuscript, we introduce the concept of a discrete dynamical system (Ⱬ,Ψ) in BCK-algebra where Ⱬ is a BCK-algebra and Ψ is a homomorphism from Ⱬ to Ⱬ and establish some of their related properties. We prove that the set of all fixed points and the set of all periodic points in BCK-algebra Ⱬ are the BCK-subalgebras. We show that when a subset of BCK-algebra Ⱬ is invariant concerning Ψ. We prove that the set of all fixed points and the set of all periodic points in commutative BCK-algebra Ⱬ with relative cancellation property are the ideals of Ⱬ. We also prove that the set of all fixed points in Ⱬ is an S-invariant subset of a BCK-algebra Ⱬ.

Keywords

• BCK-Algebra
• S-invariant or (Strongly invariant) set.
• discrete dynamical system
• periodic points
• fixed points
• invariant set
• strongly invariant

References

1. Y. Imai, K. Iseki, On Axiom Systems of Propositional Calculi 14, Proceedings of the Japan Academy 42 (1) (1966) 19-22.
2. K. Iseki, S. Tanaka, An Introduction to the Theory of BCK-Algebras, Mathematics 23 (1978) 1-26.
3. A. N Prior, Formal Logic, 2nd edition, Oxford, 1962.
4. K. Iseki, On Ideals in BCK-Algebras, Mathematics Seminar Note Kobe University 3(1) (1975) 1-12.
5. K. Iseki, Some Topics from the Category of BCK-Algebra, Mathematics Seminar Note 7 (1978) 465-468.
6. J. Meng, Y. B. Jun, BCK-Algebras, Kyung Moon Sa Co., Seoul Korea, 1994.
7. J. P. Holms, Poincare, Celestial Mechanics, Dynamical-Systems Theory and Chaos, Physics Reports 193(3) (1990) 137-163.
8. T. W. Gamelin, A History of Complex Dynamics,from Schroder to Fatou and Julia, By Daniel S. Alexander. Wiesbaden (Vieweg). Historia Mathematica, 23(1) (1996) 74-86.

9. G. D. Birkhoff, Dynamical systems, American Mathematical Society, (9), 1927.
10. K. S. Sibirskii, Introduction to Topological Dynamics, Noordhoff International Pub., 1975.
11. R. L. Devaney, Chaotic Dynamical System (second edition), Addison-Wesley, 1989.
12. E. M. Lui, Structural Stability, Structural Engineering and Geomechanics-Volume 1, 2020.
13. C. E. Silva, Invitation to Ergodic Theory, American Mathematical Society, 2008.
14. D. Dikranjan, A. G. Bruno, Discrete Dynamical Systems in Group Theory, Note di Matematica 33(1) (2013) 1-48.
15. D. Khan, A. Rehman, N. Sheikh, S. Iqbal, I. Ahmed, Properties of Discrete Dynamical System in BCI-Algebra, International Journal of Management and Fuzzy Systems 6(3) (2020) 53-58.
16. H. Yutani, Co-equalizer in the Category of BCK-algebras. Mathematics Seminar Notes 6(2) (1978) 187-188.
17. A. Dvurecenskij, On Categorical Equivalences of Commutative BCK-Algebras, An International Journal for Symbolic Logic 64(1) (2000) 21-36.
18. G. Muhiuddin, H. S. Kim, S. Z. Song, Y. B. Jun, Hesitant Fuzzy Translations and Extensions of Subalgebras and Ideals in BCK/BCI-Algebras, Journal of Intelligent and Fuzzy Systems 32(1) (2017) 43-48.
19. Y. B. Jun, S. S. Ahn, G. Muhiuddin, Hesitant Fuzzy Soft Subalgebras and Ideals in BCK/BCI-Algebras, The Scientific World Journal Article ID 763929 (2014) 7 pages.
20. Y. B. Jun, G. Muhiuddin, A. M. Al-roqi, Ideal theory of BCK/BCI-Algebras based on Double-Framed Soft Sets, Applied Mathematics & Information Sciences 7(5) (2013) 1879-1887.
21. Y. B. Jun, G. Muhiuddin, M. A. Öztürk, E. H. Roh, Cubic Soft Ideals in BCK/BCI-Algebras, Journal of Computational Analysis and Applications 22(5) (2017) 929-940.