# On the Digital Pontryagin Algebras

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. History and Hopf Space

#### 1.2. Motivation

#### 1.3. Organization of the Paper

## 2. Preliminaries

**Definition**

**1**

**.**Two points $p=({p}_{1},{p}_{2},\dots ,{p}_{n})$ and $q=({q}_{1},{q}_{2},\dots ,{q}_{n})$ with $p\ne q$ in ${\mathbb{Z}}^{n}$ are $k(u,n)$-adjacent if

- (1)
- there are at most u distinct indices i with the property $|{p}_{i}-{q}_{i}|=1$; and
- (2)
- if $|{p}_{j}-{q}_{j}|\ne 1$, then ${p}_{j}={q}_{j}$ for all indices j.

- the $k(1,1)$-adjacent points of $\mathbb{Z}$ are called 2-adjacent; and
- the $k(1,2)$-adjacent points of ${\mathbb{Z}}^{2}$ are called 4-adjacent, and the $k(2,2)$-adjacent points in ${\mathbb{Z}}^{2}$ are called 8-adjacent.

**Definition**

**2**

**.**A digital image $(X,{k}_{X})$ in ${\mathbb{Z}}^{n}$ is said to be ${k}_{X}$-connected if for every pair of points $\{x,y\}\subset X$ with $x\ne y$, there exists a set $P=\{{x}_{0},{x}_{1},\dots ,{x}_{s}\}\subset X$ of $s+1$ distinct points such that $x={x}_{0},{x}_{s}=y$, and ${x}_{i}$ and ${x}_{i+1}$ are ${k}_{X}$-adjacent for $i=0,1,\dots ,s-1$.

**Definition**

**3.**

**Definition**

**4**

**.**Let $(X,{k}_{X})$ and $(Y,{k}_{Y})$ be digital images with ${k}_{X}$-adjacent and ${k}_{Y}$-adjacent relations, respectively, and let $f,g:X\to Y$ be $({k}_{X},{k}_{Y})$-continuous functions. Suppose that there is a positive integer m and a $({k}_{X\times {[0,m]}_{\mathbb{Z}}},{k}_{Y})$-continuous function $F:X\times {[0,m]}_{\mathbb{Z}}\to Y$ such that

- $F(x,0)=f(x)$ and $F(x,m)=g(x)$ for all $x\in X$;
- the induced function ${F}_{x}:{[0,m]}_{\mathbb{Z}}\to Y,x\in X$ defined by ${F}_{x}(t)=F(x,t)$ for all $t\in {[0,m]}_{\mathbb{Z}}$ is $(2,{k}_{Y})$-continuous; and
- the induced function ${F}_{t}:X\to Y,t\in {[0,m]}_{\mathbb{Z}}$ defined by ${F}_{t}(x)=F(x,t)$ for all $x\in X$ is $({k}_{X},{k}_{Y})$-continuous.

**Definition**

**5**

**.**A pointed digital image with ${k}_{X}$-adjacent relation is a triplet $(X,{x}_{0},{k}_{X})$, where X is a digital image and ${x}_{0}\in X$. In this case, ${x}_{0}$ is said to be a base point of $(X,{x}_{0},{k}_{X})$. A pointed digital continuous function

## 3. Digital Homology Modules

- ${\u03f5}_{0}^{n}:({r}_{0},{r}_{1},\dots ,{r}_{n-1})\u27fc(0,{r}_{0},{r}_{1},\dots ,{r}_{n-1})$; and
- ${\u03f5}_{i}^{n}:({r}_{0},{r}_{1},\dots ,{r}_{n-1})\u27fc({r}_{0},\dots ,{r}_{i-1},0,{r}_{i},\dots ,{r}_{n-1})$ for $i\ge 1$,

**Definition**

**6.**

**Proposition**

**1.**

**Proof.**

## 4. Digital Hopf Spaces and Pontryagin Algebras

**Definition**

**7**

**.**Let ${e}_{{y}_{0}}:Y\to Y$ be a constant function at ${y}_{0}$ and let ${1}_{Y}:Y\to Y$ be an identity function on Y. A digital Hopf space $Y=(Y,{y}_{0},{k}_{Y},{m}_{Y})$ (sometimes denoted as $(Y,{y}_{0})$ for short) consists of a pointed digital image $(Y,{y}_{0})$ with an adjacent relation ${k}_{Y}$ and a $({k}_{Y\times Y},{k}_{Y})$-continuous function ${m}_{Y}:Y\times Y\to Y$ so that the following diagram

**Definition**

**8**

**.**Let $(Y,{y}_{0},{k}_{Y},{m}_{Y})$ be a digital Hopf space with a digital multiplication

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Definition**

**9.**

**Definition**

**10.**

**Theorem**

**3.**

**Proof.**

**Example**

**1.**

**Definition**

**11.**

**Definition**

**12**

**.**A digital multiplication ${m}_{Y}:Y\times Y\to Y$ on a digital Hopf space $(Y,{y}_{0},{k}_{Y},{m}_{Y})$ is said to be digital homotopy associative if the following diagram

**Definition**

**13**

**.**Let

**Theorem**

**4.**

**Proof.**

**Example**

**2.**

**Theorem**

**5.**

**Proof.**

**Example**

**3.**

**Remark**

**1.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- McAndrew, A.; Osborne, C. Algebraic methods for multidimensional digital topology. Proc. SPIE
**1993**, 2060, 14–25. [Google Scholar] - McAndrew, A.; Osborne, C. A survey of algebraic methods in digital topology. J. Math. Imaging Vis.
**1996**, 6, 139–159. [Google Scholar] [CrossRef] - Arslan, H.; Karaca, I.; Öztel, A. Homology groups of n-dimensinal images. Proc. XXI. Turkish Natl. Math. Sympos.
**2008**, 21, 1–13. [Google Scholar] - Boxer, L.; Karaca, I.; Öztel, A. Topological invariants in digital images. J. Math. Sci. Adv. Appl.
**2011**, 11, 109–140. [Google Scholar] - Ege, O.; Karaca, I. Digital homotopy fixed point theory. Comptes Rendus Acad. Sci. Paris Ser. I
**2015**, 353, 1029–1033. [Google Scholar] [CrossRef] - Lee, D.-W. On the digitally quasi comultiplications of digital images. Filomat
**2017**, 31, 1875–1892. [Google Scholar] [CrossRef] - James, I.M. On H-spaces and their homotopy groups. Quart. J. Math.
**1960**, 11, 161–179. [Google Scholar] [CrossRef] - Arkowitz, M.; Curjel, C.R. On maps of H-spaces. Topology
**1967**, 6, 137–148. [Google Scholar] [CrossRef] [Green Version] - Arkowitz, M.; Lupton, G. Loop-theoretic properties of H-spaces. Math. Proc. Camb. Philos. Soc.
**1991**, 110, 121–136. [Google Scholar] [CrossRef] - Ganea, T. Cogroups and suspensions. Invent. Math.
**1970**, 9, 185–197. [Google Scholar] [CrossRef] - Lee, D.-W. Phantom maps and the Gray index. Topol. Appl.
**2004**, 138, 265–275. [Google Scholar] [CrossRef] [Green Version] - Lee, D.-W. On the same n-type conjecture for the suspension of the infinite complex projective space. Proc. Am. Math. Soc.
**2009**, 137, 1161–1168. [Google Scholar] [CrossRef] - Arkowitz, M.; Lee, D.-W. Properties of comultiplications on a wedge of spheres. Topol. Appl.
**2010**, 157, 1607–1621. [Google Scholar] [CrossRef] [Green Version] - Lee, D.-W. On the same n-type structure for the suspension of the Eilenberg-MacLane spaces. J. Pure Appl. Algebra
**2010**, 214, 2027–2032. [Google Scholar] [CrossRef] - Arkowitz, M.; Lee, D.-W. Comultiplications on a wedge of two spheres. Sci. China Math.
**2011**, 54, 9–22. [Google Scholar] [CrossRef] - Lee, D.-W. On the same n-type of the suspension of the infinite quaternionic projective space. J. Pure Appl. Algebra
**2013**, 217, 1325–1334. [Google Scholar] [CrossRef] - Lee, D.-W. On the generalized same N-type conjecture. Math. Proc. Camb. Phil. Soc.
**2014**, 157, 329–344. [Google Scholar] [CrossRef] - Lee, D.-W. On the same N-types for the wedges of the Eilenberg-MacLane spaces. Chin. Ann. Math. Ser. B
**2016**, 37, 951–962. [Google Scholar] [CrossRef] - Lee, D.-W. Comultiplication structures for a wedge of spheres. Filomat
**2016**, 30, 3525–3546. [Google Scholar] [CrossRef] - Lee, D.-W. Algebraic loop structures on algebra comultiplications. Open Math.
**2019**, 17, 742–757. [Google Scholar] [CrossRef] - Lee, D.-W.; Lee, S. Homotopy comultiplications on the k-fold wedge of spheres. Topol. Appl.
**2019**, 254, 145–170. [Google Scholar] [CrossRef] - Lee, D.-W. Comultiplications on the localized spheres and Moore spaces. Mathematics
**2020**, 8, 86. [Google Scholar] [CrossRef] [Green Version] - Eckhardt, U.; Latecki, L. Digital topology. In Current Topics in Pattern Recognition Research, (Research Trends); Council of Scientific Information: Trivandrum, India, 1995; Available online: https://www.math.uni-hamburg.de/home/eckhardt/P124_941.pdf (accessed on 18 May 2020).
- Lee, D.-W. Algebraic inverses on Lie algebra comultiplications. Symmetry
**2020**, 12, 565. [Google Scholar] [CrossRef] [Green Version] - Boxer, L. Homotopy properties of sphere-like digital images. J. Math. Imaging Vis.
**2006**, 24, 167–175. [Google Scholar] [CrossRef] - Boxer, L. Digitally continuous functions. Pattern Recognit. Lett.
**1994**, 15, 833–839. [Google Scholar] [CrossRef] [Green Version] - Boxer, L. A classical construction for the digital fundamental group. J. Math. Imaging Vis.
**1999**, 10, 51–62. [Google Scholar] [CrossRef] - Rosenfeld, A. Continuous functions on digital pictures. Pattern Recognit. Lett.
**1986**, 4, 177–184. [Google Scholar] [CrossRef] - Khalimsky, E. Motion, deformation, and homotopy in finite spaces. In Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, Boston, MA, USA, 20 October 1987; pp. 227–234. [Google Scholar]
- Lee, D.-W. Digital H-spaces and actions in the pointed digital homotopy category. Appl. Algebra Eng. Comm. Comput.
**2020**, 31, 149–169. [Google Scholar] [CrossRef] - Lee, D.-W. Near-rings on digital Hopf groups. Appl. Algebra Eng. Comm. Comput.
**2018**, 29, 261–282. [Google Scholar] [CrossRef] - Lee, D.-W. Digital singular homology groups of digital images. Far East J. Math. Sci. (FJMS)
**2014**, 71, 39–63. [Google Scholar] - Vergili, T.; Karaca, I. Some properties of homology groups of Khalimsky spaces. Math. Sci. Lett.
**2015**, 4, 131–140. [Google Scholar] - Spanier, E. Algebraic Topology; McGraw-Hill: New York, NY, USA, 1996. [Google Scholar]
- Whitehead, G.W. Elements of Homotopy Theory; Graduate Texts in Math 61; Springer: New York, NY, USA; Heidelberg/Berlin, Germany, 1978. [Google Scholar]
- Ege, O.; Karaca, I. Digital H-spaces. In Proceedings of the 3rd International Symposium on Computer Science and Engineering ISCSE, Kuşadasi, Aydin, Turkey, 24–25 October 2013; pp. 133–138. [Google Scholar]
- Ege, O.; Karaca, I. Some properties of digital H-spaces. Turkish J. Electr. Eng. Comput. Sci.
**2016**, 24, 1930–1941. [Google Scholar] [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, S.; Kim, Y.; Lim, J.-E.; Lee, D.-W.
On the Digital Pontryagin Algebras. *Symmetry* **2020**, *12*, 875.
https://doi.org/10.3390/sym12060875

**AMA Style**

Lee S, Kim Y, Lim J-E, Lee D-W.
On the Digital Pontryagin Algebras. *Symmetry*. 2020; 12(6):875.
https://doi.org/10.3390/sym12060875

**Chicago/Turabian Style**

Lee, Sunyoung, Yeonjeong Kim, Jeong-Eun Lim, and Dae-Woong Lee.
2020. "On the Digital Pontryagin Algebras" *Symmetry* 12, no. 6: 875.
https://doi.org/10.3390/sym12060875