# REMARK ON GENERALIZED UNIVERSAL COVERING SPACE IN DIGITAL COVERING THEORY

• Han, Sang-Eon (Faculty of Liberal Education, Institute of Pure and Applied Mathematics, Chonbuk National University)
• Accepted : 2009.09.11
• Published : 2009.09.25
• 65 2

#### Abstract

As a survey-type article, the paper reviews the recent results on a (generalized) universal covering space in digital covering theory. The recent paper [19] established the generalized universal (2, k)-covering property which improves the universal (2, k)-covering property of [3]. In algebraic topology it is well-known that a simply connected and locally path connected covering space is a universal covering space. Unlike this property, in digital covering theory we can propose that a generalized universal covering space has its intrinsic feature. This property can be useful in classifying digital covering spaces and in studying a shortest k-path problem in data structure.

#### Keywords

digital isomorphism;digital covering;simply k-connected;universal covering property;generalized universal covering space

#### References

1. R. Ayala, E. Dominguez, A.R. Frances, and A. Quintero, Homotopy in digital spaces, Discrete Applied Math, 125(1) (2003) 3-24. https://doi.org/10.1016/S0166-218X(02)00221-4
2. 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
3. L. Boxer, Digital Products, Wedge: and Covering Spaces, Jour. of Mathematical Imaging and Vision 25 (2006) 159-171. https://doi.org/10.1007/s10851-006-9698-5
4. S.E. Han, Computer topology and its applications, Honam Math. Jour. 25(1)(2003) 153-102.
5. S.E. Han, Algorithm for discriminating digital images w.r.t. a digital ($k_0$, $k_1$)-homeomorphism, Jour. of Applied Mathematics and Computing 18(1-2)(2005) 505-512.
6. S.E. Han, Digital coverings and their applications, Jour. of Applied Mathematics and Computing 18(1-2)(2005) 487-495.
7. 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
8. S.E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27 (1)(2005) 115-129.
9. S.E. Han, Erratum to "Non-product property of the digital fundamental group", Information Sciences 176(1)(2006) 215-216. https://doi.org/10.1016/j.ins.2005.03.014
10. S.E. Han, Discrete Homotopy of a Closed k-Surface, LNCS 4040, Springer-Verlag, Berlin, pp.214-225 (2006). https://doi.org/10.1007/11774938_17
11. S.E. Han, Minimal simple closed 18-surfaces and a topological preservation of 3D surfaces, Information Sciences 176(2)(2006) 120-134. https://doi.org/10.1016/j.ins.2005.01.002
12. 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
13. 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
14. S.E. Han, Continuities and homeomorphisms in computer topology and their applications, Journal of the Korean Mathematical Society 45(4)(2008) 923-952. https://doi.org/10.4134/JKMS.2008.45.4.923
15. 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
16. S.E. Han, Map preserving local properties of a digital image Acta Applicandae Mathematicae 104(2) (2008) 177-190. https://doi.org/10.1007/s10440-008-9250-2
17. S.E. Han, The k-homotopic thinning and a torus-like digital image in $Z^n$, Journal of Mathematical Imaging and Vision 31 (1)(2008) 1-16. https://doi.org/10.1007/s10851-007-0061-2
18. S.E. Han, Cartesian product of the universal covering property Acta Applicandae Mathematicae (2009), doi 10.1007/s 10,140-008-9316-1, Online first publication. https://doi.org/10.1007/s10,140-008-9316-1
19. S.E. Han, Existence problem of a generalized universal covering space Acta Applicandae Mathematicae(2009), doi 10.1007/s 100140-008-9347-7, Online first publication. https://doi.org/10.1007/s100140-008-9347-7
20. S.E. Han, Multiplicative property of the digital fundamental group Acta Applicandae Mathematicae(2009), doi 10.1007/s 10440-009-9486-5, On line first publication. https://doi.org/10.1007/s10440-009-9486-5
21. S.E. Han, KD-($k_0$, $k_1$)-homotopy equivalence and its applications Journal of Korean Mathematical Society. to appear.
22. S.E. Han, Simple form of a digital covering space and their utility Information Sciences, submitted.
23. In-Soo Kim, S.E. Han, C.J. Yoo, The pasting property of digital continuity. Acta Applicandae Mathematicae (2009), doi 10.1007/s 10440-008-9422-0, Online first publication. https://doi.org/10.1007/s10440-008-9422-0
24. E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics(1987) 227-234.
25. T.Y. Kong, A digital fundamental group Computers and Graphics 13 (1989) 159-166. https://doi.org/10.1016/0097-8493(89)90058-7
26. T.Y. Kong, A. Rosenfeld. Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam. (1996).
27. V. Kovalevsky, Finite topology as applied to image analysis, Computer Vision, Graphics, and Image Processing 46(1989)(2) 141-161. https://doi.org/10.1016/0734-189X(89)90165-5
28. W.S. Massey, Algebraic Topology, Springer-Verlag, New York, 1977.
29. A. Rosenfeld, Digital topology, Am. Math. Mon. 86(1979) 76-87.
30. E.H. Spanier, Algebraic Topology, McGraw-Hill Inc., New York, 1966.

#### Cited by

1. COMMUTATIVE MONOID OF THE SET OF k-ISOMORPHISM CLASSES OF SIMPLE CLOSED k-SURFACES IN Z3 vol.32, pp.1, 2010, https://doi.org/10.5831/HMJ.2010.32.1.141
2. REMARKS ON DIGITAL PRODUCTS WITH NORMAL ADJACENCY RELATIONS vol.35, pp.3, 2013, https://doi.org/10.5831/HMJ.2013.35.3.515
3. 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
4. UTILITY OF DIGITAL COVERING THEORY vol.36, pp.3, 2014, https://doi.org/10.5831/HMJ.2014.36.3.695