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March  2020, 28(1): 459-469. doi: 10.3934/era.2020026

The digital smash product

Ismet Cinar , Ozgur Ege , and  Ismet Karaca

Department of Mathematics, Ege University, Bornova, 35100, Izmir, Turkey

* Corresponding author: ozgur.ege@ege.edu.tr

Received  November 2019 Published  March 2020

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.

Keywords: Digital image, digital smash product, digital shy map, digital suspension, digital cone.
Mathematics Subject Classification: Primary: 46M20; Secondary: 68U05, 68U10.
Citation: Ismet Cinar, Ozgur Ege, Ismet Karaca. The digital smash product. Electronic Research Archive, 2020, 28 (1) : 459-469. doi: 10.3934/era.2020026
References:
[1]

L. Boxer, Digitally continuous functions, Pattern Recogn. Lett., 15 (1994), 833-839.  doi: 10.1016/0167-8655(94)90012-4.  Google Scholar

[2]

L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vis., 10 (1999), 51-62.  doi: 10.1023/A:1008370600456.  Google Scholar

[3]

L. Boxer, Properties of digital homotopy, J. Math. Imaging Vis., 22 (2005), 19-26.  doi: 10.1007/s10851-005-4780-y.  Google Scholar

[4]

L. Boxer, Digital products, wedges and covering spaces, J. Math. Imaging Vis., 25 (2006), 159-171.  doi: 10.1007/s10851-006-9698-5.  Google Scholar

[5]

L. Boxer, Digital shy maps, Appl. Gen. Topol., 18 (2017), 143-152.  doi: 10.4995/agt.2017.6663.  Google Scholar

[6]

L. Boxer and I. Karaca, Fundamental groups for digital products, Adv. Appl. Math. Sci., 11 (2012), 161-179.   Google Scholar

[7]

L. Boxer, I. Karaca and A. Oztel, Topological invariants in digital images, J. Math. Sci. Adv. Appl., 11 (2011), 109-140.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. T. Herman, Oriented surfaces in digital spaces, CVGIP. Graph Model Im. Proc., 55 (1993), 381-396.  doi: 10.1006/cgip.1993.1029.  Google Scholar

[15]

I. Karaca and I. Cinar, The cohomology structure of digital Khalimsky spaces, Rom. J. Math. Comput. Sci., 8 (2018), 110-128.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[2]

L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vis., 10 (1999), 51-62.  doi: 10.1023/A:1008370600456.  Google Scholar

[3]

L. Boxer, Properties of digital homotopy, J. Math. Imaging Vis., 22 (2005), 19-26.  doi: 10.1007/s10851-005-4780-y.  Google Scholar

[4]

L. Boxer, Digital products, wedges and covering spaces, J. Math. Imaging Vis., 25 (2006), 159-171.  doi: 10.1007/s10851-006-9698-5.  Google Scholar

[5]

L. Boxer, Digital shy maps, Appl. Gen. Topol., 18 (2017), 143-152.  doi: 10.4995/agt.2017.6663.  Google Scholar

[6]

L. Boxer and I. Karaca, Fundamental groups for digital products, Adv. Appl. Math. Sci., 11 (2012), 161-179.   Google Scholar

[7]

L. Boxer, I. Karaca and A. Oztel, Topological invariants in digital images, J. Math. Sci. Adv. Appl., 11 (2011), 109-140.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. T. Herman, Oriented surfaces in digital spaces, CVGIP. Graph Model Im. Proc., 55 (1993), 381-396.  doi: 10.1006/cgip.1993.1029.  Google Scholar

[15]

I. Karaca and I. Cinar, The cohomology structure of digital Khalimsky spaces, Rom. J. Math. Comput. Sci., 8 (2018), 110-128.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Rosenfeld, Continuous functions on digital pictures, Pattern Recogn. Lett., 4 (1986), 177-184.   Google Scholar

Figure 1.  $ 2 $-adjacency in $ \mathbb{Z} $
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Figure 2.  $ 4 $ and $ 8 $ adjacencies in $ \mathbb{Z}^2 $
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Figure 3.  $ 6 $, $ 18 $ and $ 26 $ adjacencies in $ \mathbb{Z}^3 $
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Figure 4.  Digital $ 0- $sphere $ S_0 $ and digital $ 1 $-sphere $ S_1 $
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Figure 5.  $ S_1\times S_0 $ and $ S_1\wedge S_0 $
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Figure 6.  $ S_0\times I $ and $ S_0\wedge I $
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