DOI QR코드

DOI QR Code

KD-(k0, k1)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS

  • Han, Sang-Eon (FACULTY OF LIBERAL EDUCATION INSTITUTE OF PURE AND APPLIED MATHEMATICS CHONBUK NATIONAL UNIVERSITY)
  • Received : 2009.01.15
  • Published : 2010.09.01

Abstract

Let $\mathbb{Z}^n$ be the Cartesian product of the set of integers $\mathbb{Z}$ and let ($\mathbb{Z}$, T) and ($\mathbb{Z}^n$, $T^n$) be the Khalimsky line topology on $\mathbb{Z}$ and the Khalimsky product topology on $\mathbb{Z}^n$, respectively. Then for a set $X\;{\subset}\;\mathbb{Z}^n$, consider the subspace (X, $T^n_X$) induced from ($\mathbb{Z}^n$, $T^n$). Considering a k-adjacency on (X, $T^n_X$), we call it a (computer topological) space with k-adjacency and use the notation (X, k, $T^n_X$) := $X_{n,k}$. In this paper we introduce the notions of KD-($k_0$, $k_1$)-homotopy equivalence and KD-k-deformation retract and investigate a classification of (computer topological) spaces $X_{n,k}$ in terms of a KD-($k_0$, $k_1$)-homotopy equivalence.

Keywords

computer topology;digital topology;digital space;KD-($k_0$, $k_1$)-continuity;KD-k-deformation retract;digital homotopy equivalence;KD-($k_0$, $k_1$)-homotopy equivalence;KD-k-homotopic thinning

Acknowledgement

Supported by : National Foundation of Korea(NRF)

References

  1. P. Alexandroff, Diskrete Raume, Mat. Sb. 2 (1937), 501-519. https://doi.org/10.1070/SM1967v002n04ABEH002351
  2. R. Ayala, E. Dominguez, A. R. Frances, and A. Quintero, Homotopy in digital spaces, Discrete Appl. Math. 125 (2003), no. 1, 3-24. https://doi.org/10.1016/S0166-218X(02)00221-4
  3. L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839. https://doi.org/10.1016/0167-8655(94)90012-4
  4. L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vision 10 (1999), no. 1, 51-62. https://doi.org/10.1023/A:1008370600456
  5. L. Boxer, Properties of digital homotopy, J. Math. Imaging Vision 22 (2005), no. 1, 19-26. https://doi.org/10.1007/s10851-005-4780-y
  6. J. Dontchev and Haruo Maki, Groups of ${\theta}$-generalized homeomorphisms and the digital line, Topology Appl. 95 (1999), no. 2, 113-128. https://doi.org/10.1016/S0166-8641(98)00004-2
  7. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, A Compendium of Continuous Lattices, Springer-Verlag, Berlin-New York, 1980.
  8. S. E. Han, On the classification of the digital images up to a digital homotopy equivalence, The Journal of Computer and Communications Research 10 (2000), 194-207.
  9. S. E. Han, Computer topology and its applications, Honam Math. J. 25 (2003), no. 1, 153-162.
  10. S. E. Han, Comparison between digital continuity and computer continuity, Honam Math. J. 26 (2004), no. 3, 331-339.
  11. S. E. Han, Minimal digital pseudotorus with k-adjacency, k ${\in}$ 6, 18, 26, Honam Math. J. 26 (2004), no. 2, 237-246.
  12. S. E. Han, Algorithm for discriminating digital images with respect to a digital $(k_{0},\;k_{1})$-homeomorphism, Jour. of Applied Mathematics and Computing 18 (2005), no. 1-2, 505-512.
  13. S. E. Han, Non-product property of the digital fundamental group, Inform. Sci. 171 (2005), no. 1-3, 73-91. https://doi.org/10.1016/j.ins.2004.03.018
  14. S. E. Han, On the simplicial complex stemmed from a digital graph, Honam Math. J. 27 (2005), no. 1, 115-129.
  15. S. E. Han, Connected sum of digital closed surfaces, Inform. Sci. 176 (2006), no. 3, 332-348. https://doi.org/10.1016/j.ins.2004.11.003
  16. S. E. Han, Discrete Homotopy of a Closed k-Surface, LNCS 4040, Springer-Verlag, Berlin, pp. 214-225 (2006).
  17. S. E. Han, Erratum to: “Non-product property of the digital fundamental group”, Inform. Sci. 176 (2006), no. 2, 215-216. https://doi.org/10.1016/j.ins.2005.03.014
  18. S. E. Han, Remarks on digital homotopy equivalence, Honam Math. J. 29 (2007), no. 1, 101-118. https://doi.org/10.5831/HMJ.2007.29.1.101
  19. S. E. Han, Strong k-deformation retract and its applications, J. Korean Math. Soc. 44 (2007), no. 6, 1479-1503. https://doi.org/10.4134/JKMS.2007.44.6.1479
  20. S. E. Han, Continuities and homeomorphisms in computer topology and their applications, J. Korean Math. Soc. 45 (2008), no. 4, 923-952. https://doi.org/10.4134/JKMS.2008.45.4.923
  21. S. E. Han, Equivalent $(k_{0},\;k_{1})$-covering and generalized digital lifting, Inform. Sci. 178 (2008), no. 2, 550-561. https://doi.org/10.1016/j.ins.2007.02.004
  22. S. E. Han, The k-homotopic thinning and a torus-like digital image in $Z^n$, J. Math. Imaging Vision 31 (2008), no. 1, 1-16. https://doi.org/10.1007/s10851-007-0061-2
  23. S. E. Han, Map preserving local properties of a digital image, Acta Appl. Math. 104 (2008), no. 2, 177-190. https://doi.org/10.1007/s10440-008-9250-2
  24. S. E. Han, Extension of several continuities in computer topology, Bull. Korean Math. Soc., submitted.
  25. S. E. Han and N. D. Georgiou, On computer topological function space, J. Korean Math. Soc. 46 (2009), no. 4, 841-857. https://doi.org/10.4134/JKMS.2009.46.4.841
  26. S. E. Han and B. G. Park, Digital graph $(k_{0},\;k_{1})$-homotopy equivalence and its applications, http://atlas-conferences.com/c/a/k/b/35.htm (2003).
  27. E. Khalimsky, R. Kopperman, and P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology and its Applications 36 (1991), no. 1, 1-17. https://doi.org/10.1016/0166-8641(90)90031-V
  28. I.-S. Kim and S. E. Han, Digital covering theory and its applications, Honam Math. J. 30 (2008), no. 4, 589-602. https://doi.org/10.5831/HMJ.2008.30.4.589
  29. I.-S. Kim, S. E. Han, and C. J. Yoo, The almost pasting property of digital continuity, Acta Applicandae Mathematicae 110 (1) (2010), 399-408. https://doi.org/10.1007/s10440-008-9422-0
  30. T. Y. Kong, A digital fundamental group, Computers and Graphics 13 (1989), 159-166. https://doi.org/10.1016/0097-8493(89)90058-7
  31. T. Y. Kong and A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996.
  32. R. Malgouyres, Homotopy in two-dimensional digital images, Theoret. Comput. Sci. 230 (2000), no. 1-2, 221-233. https://doi.org/10.1016/S0304-3975(98)00347-8
  33. E. Melin, Extension of continuous functions in digital spaces with the Khalimsky topology, Topology Appl. 153 (2005), no. 1, 52-65. https://doi.org/10.1016/j.topol.2004.12.004
  34. A. Rosenfeld, Digital topology, Amer. Math. Monthly 86 (1979), no. 8, 621-630. https://doi.org/10.2307/2321290
  35. J. Slapal, Digital Jordan curves, Topology Appl. 153 (2006), no. 17, 3255-3264. https://doi.org/10.1016/j.topol.2005.10.011

Cited by

  1. A digitization method of subspaces of the Euclidean $$n$$ n D space associated with the Khalimsky adjacency structure vol.36, pp.1, 2017, https://doi.org/10.1007/s40314-015-0223-6
  2. PROPERTIES OF A GENERALIZED UNIVERSAL COVERING SPACE OVER A DIGITAL WEDGE vol.32, pp.3, 2010, https://doi.org/10.5831/HMJ.2010.32.3.375
  3. CATEGORY WHICH IS SUITABLE FOR STUDYING KHALIMSKY TOPOLOGICAL SPACES WITH DIGITAL CONNECTIVITY vol.33, pp.2, 2011, https://doi.org/10.5831/HMJ.2011.33.2.231
  4. SOME PROPERTIES OF LATTICE-BASED K- AND M-MAPS vol.38, pp.3, 2016, https://doi.org/10.5831/HMJ.2016.38.3.625
  5. REMARKS ON SIMPLY k-CONNECTIVITY AND k-DEFORMATION RETRACT IN DIGITAL TOPOLOGY vol.36, pp.3, 2014, https://doi.org/10.5831/HMJ.2014.36.3.519
  6. Homotopy equivalence which is suitable for studying Khalimsky nD spaces vol.159, pp.7, 2012, https://doi.org/10.1016/j.topol.2011.07.029
  7. AN EQUIVALENT PROPERTY OF A NORMAL ADJACENCY OF A DIGITAL PRODUCT vol.36, pp.1, 2014, https://doi.org/10.5831/HMJ.2014.36.1.199
  8. Digitizations associated with several types of digital topological approaches vol.36, pp.1, 2017, https://doi.org/10.1007/s40314-015-0245-0
  9. Existence of the Category DTC 2 (K) Equivalent to the Given Category KAC 2 vol.67, pp.8, 2016, https://doi.org/10.1007/s11253-016-1150-4
  10. EXTENSION PROBLEM OF SEVERAL CONTINUITIES IN COMPUTER TOPOLOGY vol.47, pp.5, 2010, https://doi.org/10.4134/BKMS.2010.47.5.915
  11. Contractibility and fixed point property: the case of Khalimsky topological spaces vol.2016, pp.1, 2016, https://doi.org/10.1186/s13663-016-0566-8
  12. DIGITAL HOMOLOGY GROUPS OF DIGITAL WEDGE SUMS vol.38, pp.4, 2016, https://doi.org/10.5831/HMJ.2016.38.4.819
  13. UTILITY OF DIGITAL COVERING THEORY vol.36, pp.3, 2014, https://doi.org/10.5831/HMJ.2014.36.3.695
  14. Homotopy based on Marcus–Wyse topology and its applications vol.201, 2016, https://doi.org/10.1016/j.topol.2015.12.047
  15. COMPARISON AMONG SEVERAL ADJACENCY PROPERTIES FOR A DIGITAL PRODUCT vol.37, pp.1, 2015, https://doi.org/10.5831/HMJ.2015.37.1.135
  16. REMARKS ON DIGITAL PRODUCTS WITH NORMAL ADJACENCY RELATIONS vol.35, pp.3, 2013, https://doi.org/10.5831/HMJ.2013.35.3.515
  17. Ultra regular covering space and its automorphism group vol.20, pp.4, 2010, https://doi.org/10.2478/v10006-010-0053-z
  18. Generalizations of continuity of maps and homeomorphisms for studying 2D digital topological spaces and their applications vol.196, 2015, https://doi.org/10.1016/j.topol.2015.05.024
  19. An MA-digitization of Hausdorff spaces by using a connectedness graph of the Marcus–Wyse topology vol.216, 2017, https://doi.org/10.1016/j.dam.2016.01.007
  20. FIXED POINT THEOREMS FOR DIGITAL IMAGES vol.37, pp.4, 2015, https://doi.org/10.5831/HMJ.2015.37.4.595
  21. COMPARISON OF CONTINUITIES IN DIGITAL TOPOLOGY vol.34, pp.3, 2012, https://doi.org/10.5831/HMJ.2012.34.3.451
  22. The fixed point property of an M -retract and its applications vol.230, 2017, https://doi.org/10.1016/j.topol.2017.08.026
  23. REMARKS ON HOMOTOPIES ASSOCIATED WITH KHALIMSKY TOPOLOGY vol.37, pp.4, 2015, https://doi.org/10.5831/HMJ.2015.37.4.577
  24. STUDY ON TOPOLOGICAL SPACES WITH THE SEMI-T½SEPARATION AXIOM vol.35, pp.4, 2013, https://doi.org/10.5831/HMJ.2013.35.4.707
  25. Homotopic properties of an MA -digitization of 2D Euclidean spaces 2017, https://doi.org/10.1016/j.jcss.2017.07.003