Algebraic structures on digital objects

Abstract

The paper aims to introduce a digital-topological ($DT$-, for brevity) $k$-ring and a $DT$-$k$-field. They are indeed endowed with both a digital image (or digital object) $(X,k)$ and a ring structure or a field structure $(X,{\ast }_1,\star )$, where $X\subset {\mathbb{Z}}^n$ and the $k$-adjacency is the digital $k$-connectivity of ${\mathbb{Z}}^n$. Besides, some properties of them are investigated. The ring $(S{C}_k^{n,l},{\ast }_1,\star )$ is proved to be isomorphic to the ring $({\mathbb{Z}}_l,+,\cdot )$, where $S{C}_k^{n,l}$ is a simple $k$-cycle with $l$ elements in ${\mathbb{Z}}^n$, $n\in \mathbb{N}\setminus \{1\}$, and $\mathbb{N}$ is the set of natural numbers. However, $(S{C}_k^{n,l},{\ast }_1,\star )$ is proved not to be a $DT$-$k$-ring. Meanwhile, we prove that for $l\in \mathcal{P}\setminus \{2,3\}$, while $(S{C}_k^{n,l},{\ast }_1,\star )$ is a field, it cannot be a $DT$-$k$-field, where $\mathcal{P}$ indicates the set of prime numbers. Besides, the paper proves that the field $(X:=\{-1,0,1\},{\ast }_1,\star )$ is a $DT$-$2$-field derived from the digital image $(X,2)$ and the field $(X:=\{-1,0,1\},{\ast }_1,\star )$, and further, $(Y:=\{0,1\},{\ast }_1,\star )$ is also a $DT$-$2$-field derived from the digital image $(Y,2)$ and the field $(Y:=\{0,1\},{\ast }_1,\star )$.

25 pages, 2 figures, original research paper