2026/07/21 by Sang-Eon Han · 1 voice
#math.GN
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, ∗1, ⋆), where X ⊂ \mathbb Zn and the k-adjacency is the digital k-connectivity of \mathbb Zn. Besides, some properties of them are investigated. The ring (SCkn,l, ∗1, ⋆) is proved to be isomorphic to the ring (\mathbb Zl, +, ⋅), where SCkn, l is a simple k-cycle with l elements in \mathbb Zn, n∈ \mathbb N∖ \1\, and \mathbb N is the set of natural numbers. However, (SCkn,l, ∗1, ⋆) is proved not to be a DT-k-ring. Meanwhile, we prove that for l ∈ P ∖ \2,3\, while (SCkn, l, ∗1, ⋆) is a field, it cannot be a DT-k-field, where P indicates the set of prime numbers. Besides, the paper proves that the field (X:=\-1, 0, 1\, ∗1, ⋆) is a DT-2-field derived from the digital image (X, 2) and the field (X:=\-1, 0, 1\, ∗1, ⋆), and further, (Y:=\0, 1\, ∗1, ⋆) is also a DT-2-field derived from the digital image (Y, 2) and the field (Y:=\0, 1\, ∗1, ⋆).