数字对象上的代数结构
Algebraic structures on digital objects
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中文总结 AI 辅助
该论文研究数字对象上的代数结构，引入DT-k环和DT-k域，研究其性质，证明了某些环和域的同构关系及是否为DT-k环或DT-k域，还给出了特定数字图像导出的DT-2域的示例。
中文摘要 AI 辅助
本文旨在引入数字拓扑（简称为DT-）k环和DT-k域。它们同时具有数字图像（或数字对象）(X,k)以及环结构或域结构(X,∗1,⋆)，其中X⊂ℤⁿ且k邻接性是ℤⁿ的数字k连通性。此外，研究了它们的一些性质。证明了环(SCₖⁿ,ₗ,∗1,⋆)与环(ℤₗ, +,·)同构，其中SCₖⁿ,ₗ是ℤⁿ中具有l个元素的简单k循环。然而，(SCₖⁿ,ₗ,∗1,⋆)不是DT-k环。同时证明对于l∈ℙ\{2,3}，当(SCₖⁿ,ₗ,∗1,⋆)是域时，它不是DT-k域。还证明了域(X:={-1,0,1},∗1,⋆)是从数字图像(X,2)导出的DT-2域，且(Y:={0,1},∗1,⋆)也是从数字图像(Y,2)导出的DT-2域。
英文摘要
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 $(Y:=\{0, 1\}, \ast_1, \star)$.
