Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

COMMUTATIVE MONOID OF THE SET OF k-ISOMORPHISM CLASSES OF SIMPLE CLOSED k-SURFACES IN Z3

  • Han, Sang-Eon (Faculty of Liberal Education, Institute of Pure and Applied Mathematics, Chonbuk National University)
  • Received : 2009.12.22
  • Accepted : 2010.02.16
  • Published : 2010.03.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we prove that with some hypothesis the set of k-isomorphism classes of simple closed k-surfaces in ${\mathbf{Z}}^3$ forms a commutative monoid with an operation derived from a digital connected sum, k ${\in}$ {18,26}. Besides, with some hypothesis the set of k-homotopy equivalence classes of closed k-surfaces in ${\mathbf{Z}}^3$ is also proved to be a commutative monoid with the above operation, k ${\in}$ {18,26}.

Keywords

References

  1. G. Bertrand and M. Malgouyres, Some topological properties of discrete surfaces, Jour. of Mathematical Imaging and Vision, 11 (1999) 207-221. https://doi.org/10.1023/A:1008348318797
  2. L. Boxer, Digitally continuous functions Pattern Recognition Letters 15 (1994) 833-839. https://doi.org/10.1016/0167-8655(94)90012-4
  3. L. Boxer, A classical construction for the digital fundamental group, Jour. of Mathematical Imaging and Vision. 10 (1999) 51-62. https://doi.org/10.1023/A:1008370600456
  4. L. Boxer, Properties of digital homotopy, Jour. of Mathematical Imaging and Vision 22 (2005) 19-26. https://doi.org/10.1007/s10851-005-4780-y
  5. A.I. Bykov, L.G. Zerkalov, M.A. Rodriguez Pineda, Index of a point of 3-D digital binary image and algorithm of computing its Euler characteristic, Pattern Recognition 32 (1999) 845-850. https://doi.org/10.1016/S0031-3203(98)00023-5
  6. L. Chen, Discrete Surfaces and Manifolds, Scientific and Practical Computing, 2004.
  7. A,V. Evako, Topological properties of closed digital spaces: One method of constructing digital models of closed continuous surfaces by using covers, Computer Vision and Image Understanding, 102 (2006) 134-144. https://doi.org/10.1016/j.cviu.2003.11.003
  8. S.E. Han, Computer topology and its applications, Honam Math. Jour., 25(1) (2003) 153-162.
  9. S.E. Han, Connected sum of digital closed surfaces, Information Sciences 116 (3)(2006) 332-348.
  10. S.E. Han, Discrete Homotopy of a Closed k-Surface, IWCIA 2006 LNCS 4040, Springer-Verlag Berlin, pp.214-225, 2006.
  11. S.E. Han, Minimal digital pseudotorus with k-adjacency, Honam Mathematical Journal 26(2) (2004) 237-246.
  12. S.E. Han, Non-product property of the digital fundamental group, Information Sciences 171 (1-3) (2005) 73-91. https://doi.org/10.1016/j.ins.2004.03.018
  13. S.E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27 (1) (2005) 115-129.
  14. S.E. Han, Minimal simple closed 18-surfeces and a topological preservation of 3D surfaces, Information Sciences 176(2)(2006) 120-134. https://doi.org/10.1016/j.ins.2005.01.002
  15. S.E. Han, Digital fundamental group and Euler characteristic of a connected sum of digital dosed surfaces, Information Sciences 177(16)(2007) 3314-3326. https://doi.org/10.1016/j.ins.2006.12.013
  16. S.E. Han, Strong k-deformation retract and its applications, Journal of the Korean Mathematical Society 44(6)(2007) 1479-1503. https://doi.org/10.4134/JKMS.2007.44.6.1479
  17. S.E. Han, Comparison among digital fundamental groups and its applications, Information Sciences 178(2008) 2091-2104 . https://doi.org/10.1016/j.ins.2007.11.030
  18. S.E. Han, Equivalent ($k_0$, $k_1$)-covering and generalized digital lifting, Information Sciences 178 (2) (2008) 550-561. https://doi.org/10.1016/j.ins.2007.02.004
  19. S.E. Han, The k-homotopic thinning and a torus-like digital image in Z", Journal of Mathematical Imaging and Vision 31 (1) (2008) 1-16. https://doi.org/10.1007/s10851-007-0061-2
  20. S.E. Han, Remark on a generalized universal covering space, Honam Mathematical Jour 31(3)(2009) 267-278. https://doi.org/10.5831/HMJ.2009.31.3.267
  21. S.E. Han, Existence problem of a generalized universal covering space, Acta Applicandae Mathematicae 109(3)(2010) 805-827. https://doi.org/10.1007/s10440-008-9347-7
  22. G. T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55 (1993) 381-396. https://doi.org/10.1006/cgip.1993.1029
  23. In-Soo Kim, S.E. Han, Digital covering theory and its application, Honam Math. Jour. 30 (4)(2008) 589-602. https://doi.org/10.5831/HMJ.2008.30.4.589
  24. In-Soo Kim, S.E. Han, C.J. Yoo, The pasting property of digital continuity, Acta Applicandae Mathematicae (2009), doi 10.1007/s10440-008-9422-0, On line first publication .
  25. R. Klette, A. Rosenfeld, Digital Geometry, Morgan Kaufmann, San Francisco, 2004
  26. T.Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996.
  27. D.G. Morgenthaler, A. Rosenfeld, Surfaces in three dimensional digital images, Information and Control, 51 (1981) 227-247. https://doi.org/10.1016/S0019-9958(81)90290-4
  28. A. Rosenfeld, Digital topology, Am. Math. Mon. 86 (1979) 76-87.