Properties of Discrete Dynamical System in BCI-Algebra

Dawood Khan; Abdul Rehman; Naveed Sheikh; Saleem Iqbal; Israr Ahmed

doi:doi:10.11648/j.ijmfs.20200603.12

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Properties of Discrete Dynamical System in BCI-Algebra

Dawood Khan, Abdul Rehman, Naveed Sheikh, Saleem Iqbal, Israr Ahmed

Received: 16 September 2020 Accepted: 13 October 2020 Published: 11 November 2020

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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.

DOI	10.11648/j.ijmfs.20200603.12
Published in	International Journal of Management and Fuzzy Systems (Volume 6, Issue 3, September 2020)
Page(s)	53-58
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

BCI-Algebra, Discrete Dynamical System, Periodic Points, Fixed Points, Invariant Set, S-invariant or (Strongly Invariant) Set

References

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[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.
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[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.

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APA Style

Dawood Khan, Abdul Rehman, Naveed Sheikh, Saleem Iqbal, Israr Ahmed. (2020). Properties of Discrete Dynamical System in BCI-Algebra. International Journal of Management and Fuzzy Systems, 6(3), 53-58. https://doi.org/10.11648/j.ijmfs.20200603.12

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ACS Style

Dawood Khan; Abdul Rehman; Naveed Sheikh; Saleem Iqbal; Israr Ahmed. Properties of Discrete Dynamical System in BCI-Algebra. Int. J. Manag. Fuzzy Syst. 2020, 6(3), 53-58. doi: 10.11648/j.ijmfs.20200603.12

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AMA Style

Dawood Khan, Abdul Rehman, Naveed Sheikh, Saleem Iqbal, Israr Ahmed. Properties of Discrete Dynamical System in BCI-Algebra. Int J Manag Fuzzy Syst. 2020;6(3):53-58. doi: 10.11648/j.ijmfs.20200603.12

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@article{10.11648/j.ijmfs.20200603.12,
  author = {Dawood Khan and Abdul Rehman and Naveed Sheikh and Saleem Iqbal and Israr Ahmed},
  title = {Properties of Discrete Dynamical System in BCI-Algebra},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {6},
  number = {3},
  pages = {53-58},
  doi = {10.11648/j.ijmfs.20200603.12},
  url = {https://doi.org/10.11648/j.ijmfs.20200603.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20200603.12},
  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.},
 year = {2020}
}

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TY  - JOUR
T1  - Properties of Discrete Dynamical System in BCI-Algebra
AU  - Dawood Khan
AU  - Abdul Rehman
AU  - Naveed Sheikh
AU  - Saleem Iqbal
AU  - Israr Ahmed
Y1  - 2020/11/11
PY  - 2020
N1  - https://doi.org/10.11648/j.ijmfs.20200603.12
DO  - 10.11648/j.ijmfs.20200603.12
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
SP  - 53
EP  - 58
PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20200603.12
AB  - 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.
VL  - 6
IS  - 3
ER  -

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Author Information

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

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APA Style

Dawood Khan, Abdul Rehman, Naveed Sheikh, Saleem Iqbal, Israr Ahmed. (2020). Properties of Discrete Dynamical System in BCI-Algebra. International Journal of Management and Fuzzy Systems, 6(3), 53-58. https://doi.org/10.11648/j.ijmfs.20200603.12

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ACS Style

Dawood Khan; Abdul Rehman; Naveed Sheikh; Saleem Iqbal; Israr Ahmed. Properties of Discrete Dynamical System in BCI-Algebra. Int. J. Manag. Fuzzy Syst. 2020, 6(3), 53-58. doi: 10.11648/j.ijmfs.20200603.12

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AMA Style

Dawood Khan, Abdul Rehman, Naveed Sheikh, Saleem Iqbal, Israr Ahmed. Properties of Discrete Dynamical System in BCI-Algebra. Int J Manag Fuzzy Syst. 2020;6(3):53-58. doi: 10.11648/j.ijmfs.20200603.12

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@article{10.11648/j.ijmfs.20200603.12,
  author = {Dawood Khan and Abdul Rehman and Naveed Sheikh and Saleem Iqbal and Israr Ahmed},
  title = {Properties of Discrete Dynamical System in BCI-Algebra},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {6},
  number = {3},
  pages = {53-58},
  doi = {10.11648/j.ijmfs.20200603.12},
  url = {https://doi.org/10.11648/j.ijmfs.20200603.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20200603.12},
  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.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Properties of Discrete Dynamical System in BCI-Algebra
AU  - Dawood Khan
AU  - Abdul Rehman
AU  - Naveed Sheikh
AU  - Saleem Iqbal
AU  - Israr Ahmed
Y1  - 2020/11/11
PY  - 2020
N1  - https://doi.org/10.11648/j.ijmfs.20200603.12
DO  - 10.11648/j.ijmfs.20200603.12
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
SP  - 53
EP  - 58
PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20200603.12
AB  - 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.
VL  - 6
IS  - 3
ER  -

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