Compatible adjacency relations for digital products

Han, Sang-Eon

doi:10.2298/fil1709787h

Cited by 4 publications

(5 citation statements)

References 27 publications

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“…Using these k-adjacency relations of Z n of (1), n ∈ N, we will call (X, k) a digital image on Z n , X ⊂ Z n . Besides, these k-adjacency relations can be essential for studying digital products with normal adjacencies [9,12] and calculating digital k-fundamental groups of digital products [9,12] (see Theorem 2 in this paper). For x, y ∈ Z with x y, the set [x, y] Z = {n ∈ Z | x ≤ n ≤ y} with 2-adjacency is called a digital interval [14,15].…”

Section: Preliminariesmentioning

confidence: 99%

“…It is called a normal adjacency of digital product [8]. Indeed, this notion is strongly related to the calculation of digital fundamental groups of digital products [9] and further, an automorphism group of a digital covering space of a digital product [9] (see Theorem 2 and Corollary 1 below).…”

Section: Remarkmentioning

confidence: 99%

“…, then we say that the k-adjacency for X 1 × X 2 is C-compatible [9]. Since both the k-fundamental group and the k-contractibility of a digital image (X, k) are so related to the study of F(Con k (X))) (see Theorem 4), we need to recall the following properties.…”

Section: Remarkmentioning

confidence: 99%

“…Theorem 2. [9] Assume that C n i ,l i k i is a simple closed k i -curve with l i elements in Z i , i ∈ {1, 2}, and they are not k i -contractible. If there is a C-compatible k-adjacency of its digital product C…”

Section: Remarkmentioning

confidence: 99%

“…The rest of the paper is organized as follows: Section 2 recalls basic backgrouds and some properties related to the study of fixed points of digital images as well as the multiplicative property of a digital fundamental group [8,9]. Section 3 explores a condition that makes F(Con k (X)) perfect.…”

Section: Introductionmentioning

confidence: 99%

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Fixed Point Sets of k-Continuous Self-Maps of m-Iterated Digital Wedges

Han

2020

Mathematics

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Let Ckn,l be a simple closed k-curves with l elements in Zn and W:=Ckn,l∨⋯∨Ckn,l︷m-times be an m-iterated digital wedges of Ckn,l, and F(Conk(W)) be an alignment of fixed point sets of W. Then, the aim of the paper is devoted to investigating various properties of F(Conk(W)). Furthermore, when proceeding with this work, this paper addresses several unsolved problems. To be specific, we firstly formulate an alignment of fixed point sets of Ckn,l, denoted by F(Conk(Ckn,l)), where l(≥7) is an odd natural number and k≠2n. Secondly, given a digital image (X,k) with X♯=n, we find a certain condition that supports n−1,n−2∈F(Conk(X)). Thirdly, after finding some features of F(Conk(W)), we develop a method of making F(Conk(W)) perfect according to the (even or odd) number l of Ckn,l. Finally, we prove that the perfectness of F(Conk(W)) is equivalent to that of F(Conk(Ckn,l)). This can play an important role in studying fixed point theory and digital curve theory. This paper only deals with k-connected digital images (X,k) such that X♯≥2.

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Section: Preliminariesmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fixed Point Sets of k-Continuous Self-Maps of m-Iterated Digital Wedges

Han

2020

Mathematics

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DT-k-group structures on digital objects and an answer to an open problem

Han

2023

Discrete Math. Algorithm. Appl.

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The paper examines various properties of [Formula: see text]-[Formula: see text]-subgroup structures and addresses an open problem on the existence of topological group structures on the [Formula: see text]-dimensional Khalimsky ([Formula: see text]-, for brevity) topological space and Marcus–Wyse ([Formula: see text]-, for short) topological plane. In particular, we obtain many types of totally [Formula: see text]-disconnected or [Formula: see text]-connected subgroups of a [Formula: see text]-connected [Formula: see text]-[Formula: see text]-group. Besides, we prove that each of the [Formula: see text]-dimensional [Formula: see text]-topological space and the [Formula: see text]-topological plane cannot be a typical topological group. Unlike an existence of a [Formula: see text]-[Formula: see text]-group structure of [Formula: see text] (see Proposition 4.7), we prove that neither of [Formula: see text] and [Formula: see text] is a topological group, where [Formula: see text] (respectively, [Formula: see text]) is a simple closed [Formula: see text]- (respectively, [Formula: see text]-) topological curve with [Formula: see text] elements in [Formula: see text] (respectively, [Formula: see text]) and the operation “∗” is a special kind of binary operation for establishing a group structure of each of [Formula: see text] and [Formula: see text]. Finally, given a [Formula: see text]-[Formula: see text]-group structure of [Formula: see text], we find several types of [Formula: see text]-[Formula: see text]-subgroup structures of it (see Theorem 5.7).

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Digitally topological groups

Han

2022

era

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<abstract><p>The purpose of the paper is to study digital topological versions of typical topological groups. In relation to this work, given a digital image $ (X, k), X \subset {\mathbb Z}^n $, we are strongly required to establish the most suitable adjacency relation in a digital product $ X \times X $, say $ G_{k^\ast} $, that supports both $ G_{k^\ast} $-connectedness of $ X \times X $ and $ (G_{k^\ast}, k) $-continuity of the multiplication $ \alpha: (X \times X, G_{k^\ast}) \to (X, k) $ for formulating a digitally topological $ k $-group (or $ DT $-$ k $-group for brevity). Thus the present paper studies two kinds of adjacency relations in a digital product such as a $ C_{k^\ast} $- and $ G_{k^\ast} $-adjacency. In particular, the $ G_{k^\ast} $-adjacency relation is a new adjacency relation in $ X \times X\subset {\mathbb Z}^{2n} $ derived from $ (X, k) $. Next, the paper initially develops two types of continuities related to the above multiplication $ \alpha $, e.g., the $ (C_{k^\ast}, k) $- and $ (G_{k^\ast}, k) $-continuity. Besides, we prove that while the $ (C_{k^\ast}, k) $-continuity implies the $ (G_{k^\ast}, k) $-continuity, the converse does not hold. Taking this approach, we define a $ DT $-$ k $-group and prove that the pair $ (SC_k^{n, l}, \ast) $ is a $ DT $-$ k $-group with a certain group operation $ \ast $, where $ SC_k^{n, l} $ is a simple closed $ k $-curve with $ l $ elements in $ {\mathbb Z}^{n} $. Also, the $ n $-dimensional digital space $ ({\mathbb Z}^n, 2n, +) $ with the usual group operation "$ + $" on $ {\mathbb Z}^n $ is a $ DT $-$ 2n $-group. Finally, the paper corrects some errors related to the earlier works in the literature.</p></abstract>

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Compatible adjacency relations for digital products

Cited by 4 publications

References 27 publications

Fixed Point Sets of k-Continuous Self-Maps of m-Iterated Digital Wedges

Fixed Point Sets of k-Continuous Self-Maps of m-Iterated Digital Wedges

DT-k-group structures on digital objects and an answer to an open problem

Digitally topological groups

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