Year 2021, Volume , Issue 35, Pages 1 - 10 2021-06-30

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

Dawood KHAN [1] , Abdul REHMAN [2]


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.
BCK-Algebra, S-invariant or (Strongly invariant) set., discrete dynamical system, periodic points, fixed points, invariant set, strongly invariant
  • Y. Imai, K. Iseki, On Axiom Systems of Propositional Calculi 14, Proceedings of the Japan Academy 42 (1) (1966) 19-22.
  • K. Iseki, S. Tanaka, An Introduction to the Theory of BCK-Algebras, Mathematics 23 (1978) 1-26.
  • A. N Prior, Formal Logic, 2nd edition, Oxford, 1962.
  • K. Iseki, On Ideals in BCK-Algebras, Mathematics Seminar Note Kobe University 3(1) (1975) 1-12.
  • K. Iseki, Some Topics from the Category of BCK-Algebra, Mathematics Seminar Note 7 (1978) 465-468.
  • J. Meng, Y. B. Jun, BCK-Algebras, Kyung Moon Sa Co., Seoul Korea, 1994.
  • J. P. Holms, Poincare, Celestial Mechanics, Dynamical-Systems Theory and Chaos, Physics Reports 193(3) (1990) 137-163.
  • 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.
  • G. D. Birkhoff, Dynamical systems, American Mathematical Society, (9), 1927.
  • K. S. Sibirskii, Introduction to Topological Dynamics, Noordhoff International Pub., 1975.
  • R. L. Devaney, Chaotic Dynamical System (second edition), Addison-Wesley, 1989.
  • E. M. Lui, Structural Stability, Structural Engineering and Geomechanics-Volume 1, 2020.
  • C. E. Silva, Invitation to Ergodic Theory, American Mathematical Society, 2008.
  • D. Dikranjan, A. G. Bruno, Discrete Dynamical Systems in Group Theory, Note di Matematica 33(1) (2013) 1-48.
  • 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.
  • H. Yutani, Co-equalizer in the Category of BCK-algebras. Mathematics Seminar Notes 6(2) (1978) 187-188.
  • A. Dvurecenskij, On Categorical Equivalences of Commutative BCK-Algebras, An International Journal for Symbolic Logic 64(1) (2000) 21-36.
  • 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.
  • 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.
  • 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.
  • 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.
Primary Language en
Subjects Mathematics
Journal Section Research Article
Authors

Orcid: 0000-0002-6850-6783
Author: Dawood KHAN (Primary Author)
Institution: University of Balochistan
Country: Pakistan


Orcid: 0000-0003-2569-8540
Author: Abdul REHMAN
Institution: University of Balochistan
Country: Pakistan


Dates

Publication Date : June 30, 2021

Bibtex @research article { jnt798677, journal = {Journal of New Theory}, issn = {2149-1402}, address = {Mathematics Department, Gaziosmanpasa University 60250 Tokat-TURKEY.}, publisher = {Gaziosmanpasa University}, year = {2021}, volume = {}, pages = {1 - 10}, doi = {10.53570/jnt.798677}, title = {A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System}, key = {cite}, author = {Khan, Dawood and Rehman, Abdul} }
APA Khan, D , Rehman, A . (2021). A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System . Journal of New Theory , (35) , 1-10 . DOI: 10.53570/jnt.798677
MLA Khan, D , Rehman, A . "A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System" . Journal of New Theory (2021 ): 1-10 <https://dergipark.org.tr/en/pub/jnt/issue/63272/798677>
Chicago Khan, D , Rehman, A . "A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System". Journal of New Theory (2021 ): 1-10
RIS TY - JOUR T1 - A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System AU - Dawood Khan , Abdul Rehman Y1 - 2021 PY - 2021 N1 - doi: 10.53570/jnt.798677 DO - 10.53570/jnt.798677 T2 - Journal of New Theory JF - Journal JO - JOR SP - 1 EP - 10 VL - IS - 35 SN - 2149-1402- M3 - doi: 10.53570/jnt.798677 UR - https://doi.org/10.53570/jnt.798677 Y2 - 2021 ER -
EndNote %0 Journal of New Theory A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System %A Dawood Khan , Abdul Rehman %T A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System %D 2021 %J Journal of New Theory %P 2149-1402- %V %N 35 %R doi: 10.53570/jnt.798677 %U 10.53570/jnt.798677
ISNAD Khan, Dawood , Rehman, Abdul . "A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System". Journal of New Theory / 35 (June 2021): 1-10 . https://doi.org/10.53570/jnt.798677
AMA Khan D , Rehman A . A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System. JNT. 2021; (35): 1-10.
Vancouver Khan D , Rehman A . A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System. Journal of New Theory. 2021; (35): 1-10.
IEEE D. Khan and A. Rehman , "A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System", Journal of New Theory, no. 35, pp. 1-10, Jun. 2021, doi:10.53570/jnt.798677