@article{Han2022, 
author = {Sang-Eon Han},
title = {Digitally topological groups},
year = {2022},
journal = {Electronic Research Archive},
volume = {30},
number = {6},
pages = {2356-2384},
keywords = {digital topological version of a topological group, DT-k-group, compatible adjacency, Ck∗-adjacency, Gk∗-adjacency, Gk∗-continuity, digital topology},
url = {https://www.sciopen.com/article/10.3934/era.2022120},
doi = {10.3934/era.2022120},
abstract = {The purpose of the paper is to study digital topological versions of typical topological groups. In relation to this work, given a digital image  (X,k),X⊂Zn, we are strongly required to establish the most suitable adjacency relation in a digital product  X×X, say  Gk∗, that supports both  Gk∗-connectedness of  X×X and  (Gk∗,k)-continuity of the multiplication  α:(X×X,Gk∗)→(X,k) for formulating a digitally topological  k-group (or  DT- k-group for brevity). Thus the present paper studies two kinds of adjacency relations in a digital product such as a  Ck∗- and  Gk∗-adjacency. In particular, the  Gk∗-adjacency relation is a new adjacency relation in  X×X⊂Z2n derived from  (X,k). Next, the paper initially develops two types of continuities related to the above multiplication  α, e.g., the  (Ck∗,k)- and  (Gk∗,k)-continuity. Besides, we prove that while the  (Ck∗,k)-continuity implies the  (Gk∗,k)-continuity, the converse does not hold. Taking this approach, we define a  DT- k-group and prove that the pair  (SCkn,l,∗) is a  DT- k-group with a certain group operation  ∗, where  SCkn,l is a simple closed  k-curve with  l elements in  Zn. Also, the  n-dimensional digital space  (Zn,2n,+) with the usual group operation " +" on  Zn is a  DT- 2n-group. Finally, the paper corrects some errors related to the earlier works in the literature.}
}