Some Properties of Lattice-Based K- And M-Maps

Han, Sang-Eon; Lee, Sik

doi:10.5831/hmj.2016.38.3.625

Cited by 3 publications

(6 citation statements)

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“…Recently, the paper [13] improved the LM A-map in [14] as follows: Let us now develop the notion of a generalized LM A-map as follows: It turns out that [13] this version is both a kind of a generalization of an LM A-map in [14] and an improved and corrected version of an LM A-map in [14]. Thus the LM A-map of the paper [14] can be replaced by the current generalized LM A-map.…”

Section: Summary and Further Workmentioning

confidence: 99%

“…Thus the LM A-map of the paper [14] can be replaced by the current generalized LM A-map. Hereafter, we will call the map F in Definition 20 an LM A-map instead of a generalized LM A-map [13]. Besides, the paper [13] also improved the LA-map in [12] as follows: …”

Section: Summary and Further Workmentioning

confidence: 99%

“…Hereafter, we will call the map F in Definition 21 an LA-map instead of a generalized LA-map [13]. Thus the LA-map of the paper [12] can be replaced by the current generalized LA-map.…”

Section: Summary and Further Workmentioning

confidence: 99%

“…Denition. [13] Consider the map F : (X, E n X ) → (Y, E n Y ) such that D KA (F ) := f : D KA (X) → D KA (Y ) is an A-map, where D KA (F ) := f is induced by F satisfying that for any point p ∈ D KA (X)      F (N K (p) ∩ X) ⊂ N K (f (p)) ∩ Y and f maps p to q i , where…”

Section: Summary and Further Workmentioning

confidence: 99%

“…It turns out that [13] this version is both a kind of a generalization of an LAmap in [12] and an improved and corrected version of an LA-map in [12]. Hereafter, we will call the map F in Denition 21 an LA-map instead of a generalized LA-map [13]. Thus the LA-map of the paper [12] can be replaced by the current generalized LA-map.…”

Section: Summary and Further Workmentioning

confidence: 99%

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U(k)- AND L(k)-HOMOTOPIC PROPERTIES OF DIGITIZATIONS OF nD HAUSDORFF SPACES

Han¹

2017

HJMS

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Abstract. For X ⊂ R n let (X, E n X ) be the usual topological space induced by the nD Euclidean topological space (R n , E n ). Based on the upper limit (U -, for short) topology (resp. the lower limit (L-, for brevity) topology), after proceeding with a digitization of (X,) with a digital k-connnectivity, we obtain a digital image from the viewpoint of digital topology in a graph-theoretical approach, i.e. Rosenfeld model [25], denoted by D U (k) (X) (resp. D L(k) (X)) in the present paper. Since a Euclidean topological homotopy has some limitations of studying a digitization of (X, E n X ), the present paper establishes the so called U (k)-homotopy (resp. L(k)-homotopy) which can be used to study homotopic properties of both (X, E n X ) and D U (k) (X) (resp. both (X, E n X ) and D L(k) (X)). The goal of the paper is to study some relationships among an ordinary homotopy equivalence, a U (k)-homotopy equivalence, an L(k)-homotopy equivalence and a khomotopy equivalence. Finally, we classify (X, E n X ) in terms of a U (k)-homotopy equivalence and an L(k)-homotopy equivalence. This approach can be used to study applied topology, approximation theory and digital geometry.

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Section: Summary and Further Workmentioning

confidence: 99%

Section: Summary and Further Workmentioning

confidence: 99%

“…Hereafter, we will call the map F in Definition 21 an LA-map instead of a generalized LA-map [13]. Thus the LA-map of the paper [12] can be replaced by the current generalized LA-map.…”

Section: Summary and Further Workmentioning

confidence: 99%

Section: Summary and Further Workmentioning

confidence: 99%

Section: Summary and Further Workmentioning

confidence: 99%

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U(k)- AND L(k)-HOMOTOPIC PROPERTIES OF DIGITIZATIONS OF nD HAUSDORFF SPACES

Han¹

2017

HJMS

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Homotopic properties of an MA-digitization of 2D Euclidean spaces

Han

2018

Journal of Computer and System Sciences

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Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces

Han

2020

Hacettepe Journal of Mathematics and Statistics

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For X(⊂ R n ), assume the subspace (X, E n X ) induced by the n-dimensional Euclidean topological space (R n , E n ). Let Z be the set of integers. Khalimsky topology on Z, denoted by (Z, κ), is generated by the set {{2m − 1, 2m, 2m + 1} | m ∈ Z} as a subbase. Besides, Khalimsky topology on Z n , n ∈ N, denoted by (Z n , κ n ), is a product topology induced by (Z, κ). Proceeding with a digitization of (X, E n X ) in terms of the Khalimsky (K-, for short) topology, we obtain a K-digitized space in Z n , denoted by D K (X)(⊂ Z n ), which is a K-topological space. Considering further D K (X) with K-adjacency, we obtain a topological graph related to the K-topology (a KA-space for short) denoted by D KA (X) (see an algorithm in Section 3). Motivated by an A-homotopy between A-maps for KA-spaces, the present paper establishes a new homotopy, called an LA-homotopy, which is suitable for studying homotopic properties of both (X, E n X ) and D KA (X) because a homotopy for Euclidean topological spaces has some limitations of digitizing (X, E n X ). The goal of the paper is to study some relationships among an ordinary homotopy equivalence for spaces (X, E n X ), an LA-homotopy equivalence for spaces (X, E n X ), and an A-homotopy equivalence for KA-spaces D KA (X). Finally, we classify KA-spaces (resp. (X, E n X )) via an A-homotopy equivalence (resp. an LA-homotopy equivalence). This approach can facilitate studies of applied topology, approximation theory and digital geometry.

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Some Properties of Lattice-Based K- And M-Maps

Abstract: Abstract. The recent papers [7,12] developed two maps named by an LA- [7] and an LM A-map [12]. The present paper studies their properties and further, develops generalized versions of an LA-and an LM A-map, which makes the LA- [7] and the LM A-map [12] improved.

Cited by 3 publications

References 14 publications

U(k)- AND L(k)-HOMOTOPIC PROPERTIES OF DIGITIZATIONS OF nD HAUSDORFF SPACES

U(k)- AND L(k)-HOMOTOPIC PROPERTIES OF DIGITIZATIONS OF nD HAUSDORFF SPACES

Homotopic properties of an MA-digitization of 2D Euclidean spaces

Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces

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