Volume 35, Issue 1 (2026)

Exploring the Containment Hierarchy of Subrings over Rings ${\mathbb{Z}}_2$ to ${\mathbb{Z}}_9$ in Two-Dimensional Cellular Automata

Velvet Alexander
School of Advanced Sciences, Vellore Institute of Technology
Chennai Campus, Tamil Nadu, India—600127
Ummity Srinivasa Rao
Kedar Shinde
Jayaram Balabaskaran
School of Computer Science and Engineering
Vellore Institute of Technology
Chennai Campus, Tamil Nadu, India—600127
R. J. Hari
Indian Institute of Information Technology Kottayam
Valavoor P.O, Pala
Kottayam, Kerala, India—686635
Asif Islam
Aqdus Bin Bari
Zakir Husain College of Engineering and Technology
Aligarh Muslim University, Uttar Pradesh, India—202002

Abstract

This paper investigates the structural properties of two-dimensional cellular automata (2DCAs) over rings ${\mathbb{Z}}_2$ to ${\mathbb{Z}}_9$, focusing on rule composition, subring hierarchy and linear evolution. We prove that rule composition is commutative across all rings and that matrix transformations preserve hierarchical relationships. The subring containment is determined by the divisors of n: prime rings have only trivial subrings, while composite rings exhibit structured hierarchies. Let $I\in {\mathbb{Z}}_k^{n⨯n}$ be a configuration matrix. For rings in ${\mathfrak{S}}_1=\left\{{\mathbb{Z}}_2,{\mathbb{Z}}_4,{\mathbb{Z}}_8\right\}$, repeated elementwise addition modulo k maps the entries of I to their immediate subrings in the containment hierarchy. For ${\mathfrak{S}}_2=\left\{{\mathbb{Z}}_3,{\mathbb{Z}}_9\right\}$, triple summation modulo k similarly restricts the values to their corresponding subrings. Furthermore, we extend Moore neighborhood-based two-dimensional cellular automaton (2DCA) rules to rings from ${\mathbb{Z}}_2$ to ${\mathbb{Z}}_5$ and ${\mathbb{Z}}_7$ to ${\mathbb{Z}}_9$, proving that at time step t, rule matrices generate multiple non-overlapping replicas of the initial configuration across each ring and its subrings.
        Experiments in C++ across various rings and image sizes revealed two key patterns requiring further mathematical explanation. Rings with only trivial subrings—${\mathbb{Z}}_2$, ${\mathbb{Z}}_3$, ${\mathbb{Z}}_5$ and ${\mathbb{Z}}_7$—replicate the initial image at time steps $t=p^k$, where $p\in \left\{2,3,5,7\right\}$ and $k\ge 0$. In contrast, rings with nontrivial subrings—${\mathbb{Z}}_4$ and ${\mathbb{Z}}_8$—generate multiple replicas at $t=2^k$, while ${\mathbb{Z}}_9$ exhibits replication at $t=3^k$, producing both the original image and its subrings. The ring ${\mathbb{Z}}_6$, with disjoint subrings and no clear hierarchy, shows no such structured replication.

Keywords: two-dimensional cellular automata; rings; subrings; Moore neighborhood; linear rules; ${\mathbb{Z}}_2$ to ${\mathbb{Z}}_9$; image processing

Cite this publication as:
V. Alexander, U. S. Rao, K. Shinde, J. Balabaskaran, R. J. Hari, A. Islam and A. B. Bari, “Exploring the Containment Hierarchy of Subrings over Rings ${\mathbb{Z}}_2$ to ${\mathbb{Z}}_9$ in Two-Dimensional Cellular Automata,” Complex Systems, 35(1), 2026 pp. 11–38.
https://doi.org/10.25088/ComplexSystems.35.1.11