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The Most Refined Axiom for a Digital Covering Space and Its Utilities

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  • Sang-Eon Han

    (Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk 54896, Korea)

Abstract

This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local ( k 0 , k 1 ) -isomorphisms and use the most simplified local ( k 0 , k 1 ) -isomorphism. This approach is indeed a key step to make the axioms for a digital covering space very refined. In this paper, the most refined local ( k 0 , k 1 ) -isomorphism is proved to be a ( k 0 , k 1 ) -covering map, which implies that the earlier axioms for a digital covering space are significantly simplified with one axiom. This finding facilitates the calculations of digital fundamental groups of digital images using the unique lifting property and the homotopy lifting theorem. In addition, consider a simple closed k : = k ( t , n ) -curve with five elements in Z n , denoted by S C k n , 5 . After introducing the notion of digital topological imbedding, we investigate some properties of S C k n , 5 , where k : = k ( t , n ) , 3 ≤ t ≤ n . Since S C k n , 5 is the minimal and simple closed k -curve with odd elements in Z n which is not k -contractible, we strongly study some properties of it associated with generalized digital wedges from the viewpoint of fixed point theory. Finally, after introducing the notion of generalized digital wedge, we further address some issues which remain open. The present paper only deals with k -connected digital images.

Suggested Citation

  • Sang-Eon Han, 2020. "The Most Refined Axiom for a Digital Covering Space and Its Utilities," Mathematics, MDPI, vol. 8(11), pages 1-21, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:1868-:d:435376
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    References listed on IDEAS

    as
    1. Efim Khalimsky & Ralph Kopperman & Paul R. Meyer, 1990. "Boundaries in digital planes," International Journal of Stochastic Analysis, Hindawi, vol. 3, pages 1-29, January.
    2. Han, Sang-Eon, 2019. "Estimation of the complexity of a digital image from the viewpoint of fixed point theory," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 236-248.
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    Cited by:

    1. Sang-Eon Han, 2021. "Discrete Group Actions on Digital Objects and Fixed Point Sets by Iso k (·)-Actions," Mathematics, MDPI, vol. 9(3), pages 1-25, February.

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