# ALGEBRAIC STRUCTURES ON A SET OF DISCRETE DYNAMICAL SYSTEM AND A SET OF PROFILE

### Abstract

*A discrete dynamical system is represented as a directed graph with graph nodes called states that can be seen on the dynamical map. This discrete dynamical system is symbolized by * *, where * * is a finite set of states and the function g* * is a function from * * to * *. In the dynamical map, the discrete dynamical system has a height where the number of states in each height is called a profile. The set of discrete dynamical systems has an addition operation defined as a disjoint union on the graph and a multiplication operation defined as a tensor product on the graph. The set of discrete dynamical systems and the set of profiles are very interesting to observe from the algebraic point of view. Considering operation on the set of discrete dynamical systems and the set of profiles, we can see their algebraic structure. By recognizing the algebraic structure, it will be easy to solve the polynomial equation in the discrete dynamical system and in the profile.* *In this research, we will investigate the algebraic structure of discrete dynamical systems and the set of profiles. This research shows that the set of discrete dynamical system has an algebraic structure, which is a commutative semiring and the set of profiles has an algebraic structure, which is a commutative semiring and * *-semimodule. Moreover, both sets have the same property, which is isomorphic to the set of non-negative integers.*

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### References

Hartrisari, “Sistem dinamik: konsep sistem dan permodelan untuk industri dan lingkungan,” SEAMEO Biotrop, 2007.

A. Dennunzio, V. Dorigatti, E. Formenti, L. Manzoni, and A. E. Porreca, Polynomial Equations over Finite, Discrete-Time Dynamical Systems, vol. 11115 LNCS. Springer International Publishing, 2018. doi: 10.1007/978-3-319-99813-8_27.

L. AlSuwaidan and M. Ykhlef, “Toward Information Diffusion Model for Viral Marketing in Business,” 2016.

F. W. Lawvere and S. H. Schanuel, “Conceptual mathematics: a first introduction to categories,” Choice Rev. Online, vol. 36, no. 02, pp. 36-1018-36–1018, 2009, doi: 10.5860/choice.36-1018.

J. Á. Cid and J. Mawhin, “The Brouwer fixed point theorem and periodic solutions of differential equations,” J. Fixed Point Theory Appl., vol. 25, no. 1, pp. 1–12, 2023, doi: 10.1007/s11784-022-01023-x.

C. Gaze-maillot and A. E. Porreca, “Profiles of dynamical systems and their algebra,” arXiv2008.00843 [cs, math], 2020, [Online]. Available: http://arxiv.org/abs/2008.00843

M. L. Kilpack, “Algebraic Structure Representations for Lattices,” arXiv Prepr. arXiv, 2018.

A. Triyani and S. A. Larasati, “Algebraic Structures of Interval Sets,” 2020.

J. Liesen, Algebraic Structures. 2015.

J. R. Guzmán, “On a Lie algebraic Structure Associated with a Non-Linear Dynamical System,” arXiv Prepr., 2019.

E. Patterson, “The algebra and machine representation of statistical models,” no. June, 2020, [Online]. Available: http://arxiv.org/abs/2006.08945

G. G. de Castro and E. J. Kang, “Boundary path groupoids of generalized Boolean dynamical systems and their C⁎-algebras,” J. Math. Anal. Appl., vol. 518, no. 1, 2023, doi: 10.1016/j.jmaa.2022.126662.

I. Qaralleh and F. Mukhamedov, “Volterra evolution algebras and their graphs,” Linear Multilinear Algebr., vol. 69, no. 12, pp. 2228–2244, 2021, doi: 10.1080/03081087.2019.1664387.

J. Tomiyama, “Hulls and kernels from topological dynamical systems and their applications to homeomorphism C*-algebras,” J. Math. Soc. Japan, vol. 56, no. 2, pp. 349–364, 2004, doi: 10.2969/jmsj/1191418634.

J. A. Aledo, S. Martinez, and J. C. Valverde, “Parallel dynamical systems over graphs and related topics: A survey,” J. Appl. Math., vol. 2015, pp. 26–29, 2015, doi: 10.1155/2015/594294.

D. Khan, A. Rehman, N. Sheikh, S. Iqbal, and I. Ahmed, “Properties of Discrete Dynamical System in BCI-Algebra,” Int. J. Manag. Fuzzy Syst., vol. 6, no. 3, p. 53, 2020, doi: 10.11648/j.ijmfs.20200603.12.

D. Khan and A. Rehman, “A New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System,” J. New Theory, vol. 52, no. 3, pp. 95–100, 2021.

K. H. Rosen, Handbook of Discrete and Combinatorial Mathematics. CRC press, 1999.

*BAREKENG: J. Math. & App.*, vol. 18, no. 1, pp. 0065-0074, Mar. 2024.

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