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The digital smash product
Department of Mathematics, Ege University, Bornova, 35100, Izmir, Turkey |
In this paper, we construct the smash product from the digital viewpoint and prove some its properties such as associativity, distributivity, and commutativity. Moreover, we present the digital suspension and the digital cone for an arbitrary digital image and give some examples of these new concepts.
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[1] |
L. Boxer,
Digitally continuous functions, Pattern Recogn. Lett., 15 (1994), 833-839.
doi: 10.1016/0167-8655(94)90012-4. |
[2] |
L. Boxer,
A classical construction for the digital fundamental group, J. Math. Imaging Vis., 10 (1999), 51-62.
doi: 10.1023/A:1008370600456. |
[3] |
L. Boxer,
Properties of digital homotopy, J. Math. Imaging Vis., 22 (2005), 19-26.
doi: 10.1007/s10851-005-4780-y. |
[4] |
L. Boxer,
Digital products, wedges and covering spaces, J. Math. Imaging Vis., 25 (2006), 159-171.
doi: 10.1007/s10851-006-9698-5. |
[5] |
L. Boxer,
Digital shy maps, Appl. Gen. Topol., 18 (2017), 143-152.
doi: 10.4995/agt.2017.6663. |
[6] |
L. Boxer and I. Karaca,
Fundamental groups for digital products, Adv. Appl. Math. Sci., 11 (2012), 161-179.
|
[7] |
L. Boxer, I. Karaca and A. Oztel,
Topological invariants in digital images, J. Math. Sci. Adv. Appl., 11 (2011), 109-140.
|
[8] |
I. Cinar and I. Karaca,
Some new results on connected sum of certain digital surfaces, Malaya J. Matematik, 7 (2019), 318-325.
doi: 10.26637/MJM0702/0027. |
[9] |
O. Ege and I. Karaca, Fundamental properties of simplicial homology groups for digital images, Amer. J. Comput. Tech. Appl., 1 (2013), 25-42. Google Scholar |
[10] |
O. Ege and I. Karaca, Cohomology theory for digital images, Rom. J. Inf. Sci. Tech., 16 (2013), 10-28. Google Scholar |
[11] |
O. Ege and I. Karaca,
Digital cohomology operations, Appl. Math. Inform. Sci., 9 (2015), 1953-1960.
doi: 10.12785/amis. |
[12] |
O. Ege and I. Karaca,
Digital fibrations, Proc. Natl. Acad. Sci. India Sect. A Phys. Sci., 87 (2017), 109-114.
doi: 10.1007/s40010-016-0302-0. |
[13] |
O. Ege, I. Karaca and M. Erden Ege,
Relative homology groups of digital images, Appl. Math. Inform. Sci., 8 (2014), 2337-2345.
doi: 10.12785/amis/080529. |
[14] |
G. T. Herman,
Oriented surfaces in digital spaces, CVGIP. Graph Model Im. Proc., 55 (1993), 381-396.
doi: 10.1006/cgip.1993.1029. |
[15] |
I. Karaca and I. Cinar,
The cohomology structure of digital Khalimsky spaces, Rom. J. Math. Comput. Sci., 8 (2018), 110-128.
|
[16] |
I. Karaca and O. Ege, Some results on simplicial homology groups of 2D digital images, Int. J. Inform. Comput. Sci., 1 (2012), 198-203. Google Scholar |
[17] |
T. Y. Kong, R. Kopperman and P. R. Meyer,
A topological approach to digital topology, Amer. Math. Monthly, 98 (1991), 901-917.
doi: 10.1080/00029890.1991.12000810. |
[18] |
T. Y. Kong and A. Rosenfeld, Digital topology, introduction and survey, Comput. Vision Graphics Image Process., 48 (1989), 357-393. Google Scholar |
[19] |
R. Kopperman, P. R. Meyer and R. G. Wilson,
A Jordan surface theorem for three dimensional digital spaces, Discrete Comput. Geom., 6 (1991), 155-161.
doi: 10.1007/BF02574681. |
[20] |
V. A. Kovalevsky, Finite topology as applied to image analysis, Comput. Vision Graphics Image Process., 46 (1989), 141-161. Google Scholar |
[21] |
A. Rosenfeld,
Digital topology, Amer. Math. Monthly, 86 (1979), 621-630.
doi: 10.1080/00029890.1979.11994873. |
[22] |
A. Rosenfeld, Continuous functions on digital pictures, Pattern Recogn. Lett., 4 (1986), 177-184. Google Scholar |
show all references
References:
[1] |
L. Boxer,
Digitally continuous functions, Pattern Recogn. Lett., 15 (1994), 833-839.
doi: 10.1016/0167-8655(94)90012-4. |
[2] |
L. Boxer,
A classical construction for the digital fundamental group, J. Math. Imaging Vis., 10 (1999), 51-62.
doi: 10.1023/A:1008370600456. |
[3] |
L. Boxer,
Properties of digital homotopy, J. Math. Imaging Vis., 22 (2005), 19-26.
doi: 10.1007/s10851-005-4780-y. |
[4] |
L. Boxer,
Digital products, wedges and covering spaces, J. Math. Imaging Vis., 25 (2006), 159-171.
doi: 10.1007/s10851-006-9698-5. |
[5] |
L. Boxer,
Digital shy maps, Appl. Gen. Topol., 18 (2017), 143-152.
doi: 10.4995/agt.2017.6663. |
[6] |
L. Boxer and I. Karaca,
Fundamental groups for digital products, Adv. Appl. Math. Sci., 11 (2012), 161-179.
|
[7] |
L. Boxer, I. Karaca and A. Oztel,
Topological invariants in digital images, J. Math. Sci. Adv. Appl., 11 (2011), 109-140.
|
[8] |
I. Cinar and I. Karaca,
Some new results on connected sum of certain digital surfaces, Malaya J. Matematik, 7 (2019), 318-325.
doi: 10.26637/MJM0702/0027. |
[9] |
O. Ege and I. Karaca, Fundamental properties of simplicial homology groups for digital images, Amer. J. Comput. Tech. Appl., 1 (2013), 25-42. Google Scholar |
[10] |
O. Ege and I. Karaca, Cohomology theory for digital images, Rom. J. Inf. Sci. Tech., 16 (2013), 10-28. Google Scholar |
[11] |
O. Ege and I. Karaca,
Digital cohomology operations, Appl. Math. Inform. Sci., 9 (2015), 1953-1960.
doi: 10.12785/amis. |
[12] |
O. Ege and I. Karaca,
Digital fibrations, Proc. Natl. Acad. Sci. India Sect. A Phys. Sci., 87 (2017), 109-114.
doi: 10.1007/s40010-016-0302-0. |
[13] |
O. Ege, I. Karaca and M. Erden Ege,
Relative homology groups of digital images, Appl. Math. Inform. Sci., 8 (2014), 2337-2345.
doi: 10.12785/amis/080529. |
[14] |
G. T. Herman,
Oriented surfaces in digital spaces, CVGIP. Graph Model Im. Proc., 55 (1993), 381-396.
doi: 10.1006/cgip.1993.1029. |
[15] |
I. Karaca and I. Cinar,
The cohomology structure of digital Khalimsky spaces, Rom. J. Math. Comput. Sci., 8 (2018), 110-128.
|
[16] |
I. Karaca and O. Ege, Some results on simplicial homology groups of 2D digital images, Int. J. Inform. Comput. Sci., 1 (2012), 198-203. Google Scholar |
[17] |
T. Y. Kong, R. Kopperman and P. R. Meyer,
A topological approach to digital topology, Amer. Math. Monthly, 98 (1991), 901-917.
doi: 10.1080/00029890.1991.12000810. |
[18] |
T. Y. Kong and A. Rosenfeld, Digital topology, introduction and survey, Comput. Vision Graphics Image Process., 48 (1989), 357-393. Google Scholar |
[19] |
R. Kopperman, P. R. Meyer and R. G. Wilson,
A Jordan surface theorem for three dimensional digital spaces, Discrete Comput. Geom., 6 (1991), 155-161.
doi: 10.1007/BF02574681. |
[20] |
V. A. Kovalevsky, Finite topology as applied to image analysis, Comput. Vision Graphics Image Process., 46 (1989), 141-161. Google Scholar |
[21] |
A. Rosenfeld,
Digital topology, Amer. Math. Monthly, 86 (1979), 621-630.
doi: 10.1080/00029890.1979.11994873. |
[22] |
A. Rosenfeld, Continuous functions on digital pictures, Pattern Recogn. Lett., 4 (1986), 177-184. Google Scholar |
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