Volume 6, Issue 3, September 2020, Page: 53-58
Properties of Discrete Dynamical System in BCI-Algebra
Dawood Khan, Department of Mathematics, University of Balochistan, Quetta, Pakistan
Abdul Rehman, Department of Mathematics, University of Balochistan, Quetta, Pakistan
Naveed Sheikh, Department of Mathematics, University of Balochistan, Quetta, Pakistan
Saleem Iqbal, Department of Mathematics, University of Balochistan, Quetta, Pakistan
Israr Ahmed, Department of Mathematics, University of Balochistan, Quetta, Pakistan
Received: Sep. 16, 2020;       Accepted: Oct. 13, 2020;       Published: Nov. 11, 2020
DOI: 10.11648/j.ijmfs.20200603.12      View  41      Downloads  9
Abstract
In the present manuscript we introduce the concept of some notions such as fixed points, periodic points, invariant set and strongly invariant or S-invariant set of discrete dynamical system (Z, Ψ) in BCI-algebra where in (Z, Ψ), Z is a non-empty set and supposed to be a BCI-algebra and the mapping Ψ is a homomorphism from Z to Z and establish some new homomorphic properties of BCI-algebra based on these notions. We also prove some new results related to the set that contains the all fixed points and to the set that contains all periodic points in Z such that we prove that the set of all fixed points and the set of all periodic points in BCI-algebra Z are the BCI-sub algebras. We show that when a sub set of BCI-algebra Z is an invariant set with respect to Ψ. We prove that the set of all fixed points and the set of all periodic points in p-semisimple BCI-algebra Z are the ideals of Z. We also prove that the set of all fixed points in Z is an S-invariant subset of a BCI-algebra Z. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of a discrete dynamical system in BCI-algebra and their applications in other disciplines of algebra.
Keywords
BCI-Algebra, Discrete Dynamical System, Periodic Points, Fixed Points, Invariant Set, S-invariant or (Strongly Invariant) Set
To cite this article
Dawood Khan, Abdul Rehman, Naveed Sheikh, Saleem Iqbal, Israr Ahmed, Properties of Discrete Dynamical System in BCI-Algebra, International Journal of Management and Fuzzy Systems. Vol. 6, No. 3, 2020, pp. 53-58. doi: 10.11648/j.ijmfs.20200603.12
Copyright
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
Fernandes, S., Ramos, C., Thapa, G., Lopes, L., & Grácio, C. (2018). Discrete Dynamical Systems: A Brief Survey. Journal of the Institute of Engineering, 14 (1), 35-51.
[2]
Bunder, M. W. (1998). Cancellation laws for BCI-algebra, atoms and p-semisimple BCI-algebras.". scientiae mathematicae japonicae, 1 (1), 19-22.
[3]
Devaney, R. L. (1989). chaotic dynamical systems (second edition). Addison-wesley.
[4]
Dardano, U., Dikranjan, D., & Rinauro, S. (2017). Inertial properties in groups. arXiv preprint arXiv: 1705.02954.
[5]
Dikranjan, D., & Bruno, A. G. (2013). Discrete Dynamical systems in group theory. arXiv preprint arXiv: 1308.4035.
[6]
Gamelin, T. W. (1996). A History of Complex Dynamics, from Schroder to Fatou and Julia. By Daniel S. Alexander. Wiesbaden (Vieweg). Historia Mathematica, 23 (1), 74-86.
[7]
Holms, J. P. (1990). Poincare, celestial mechanics, dynamical-systems theory and chaos. PhysicsReports 193.3, 137-163.
[8]
Hoo, C. S. (1988). A survey of BCK and BCI-algebras. Southeast Asian Bull. Math., 12.
[9]
Iseki, K. (1966). An algebra related with a propositional calculus. proc. japan. Acad 42, 26-29.
[10]
Iseki, k. (1980). On BCI-algebras. Math. Semi notes (presently Kobe. J. Math.) 8.
[11]
Iseki, K., & Tanaka, S. (1978). An introduction to the theory of BCK-algebras. Math. japonica, 23, 1-26.
[12]
Meng, J., & Jun, B. Y. (1994). BCK-algebras. Kyung Moon Sa Co., Seoul, Korea.
[13]
Meng, J., & Liu, Y. L. (2001). An introduction to BCI-algebras. (Chinese), Shanxi Scientific and Technological press, Xi'an, chaina.
[14]
Lei, T., & Xi, C. (1985). p-Radical in BCI-algebras. Math. Japonica 30, 511-517.
[15]
Lui, E. M. (2020). Structural Stability. Structural Engineering and Geomechanics-Volume 1, 198.
[16]
Prior, A. N. (1962). Formal Logic, second edition. Oxford.
[17]
De Vries, J. (2013). Elements of topological dynamics (Vol. 257). Springer Science & Business Media.
[18]
Silva, C. E. (2007). Invitation to ergodic theory (Vol. 42). American Mathematical Society.
[19]
Smale, S. (1960). Morse inequalities for dynamical system. Bulletin of the American Mathematical society, 66 (1), 43-49.
[20]
Dikranjan, D., & Bruno, A. G. (2016). Entropy on abelian groups. Advances in Mathematics, 298, 612-653.
Browse journals by subject