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 $(SC_k^{n,l}, \ast_1, \star)$ is proved to be isomorphic to the ring $({\mathbb Z}_l, +, \cdot)$, where $SC_k^{n, l}$ 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, $(SC_k^{n,l}, \ast_1, \star)$ is proved not to be a $DT$-$k$-ring. Meanwhile, we prove that for $l \in \mathcal{P} \setminus \{2,3\}$, while $(SC_k^{n, l}, \ast_1, \star)$ 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 $(Y:=\{0, 1\}, \ast_1, \star)$.