Digitally topological groups

The purpose of the paper is to study digital topological versions of typical topological groups. In relation to this work, given a digital image $(X,k),X\subset {\mathbb{Z}}^n$, we are strongly required to establish the most suitable adjacency relation in a digital product $X\times X$, say $G_{k^{\ast }}$, that supports both $G_{k^{\ast }}$-connectedness of $X\times X$ and $(G_{k^{\ast }},k)$-continuity of the multiplication $\alpha :(X\times X,G_{k^{\ast }})\to (X,k)$ 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 $C_{k^{\ast }}$- and $G_{k^{\ast }}$-adjacency. In particular, the $G_{k^{\ast }}$-adjacency relation is a new adjacency relation in $X\times X\subset {\mathbb{Z}}^{2n}$ derived from $(X,k)$. Next, the paper initially develops two types of continuities related to the above multiplication $\alpha$, e.g., the $(C_{k^{\ast }},k)$- and $(G_{k^{\ast }},k)$-continuity. Besides, we prove that while the $(C_{k^{\ast }},k)$-continuity implies the $(G_{k^{\ast }},k)$-continuity, the converse does not hold. Taking this approach, we define a $DT$- $k$-group and prove that the pair $(S{C}_k^{n,l},\ast )$ is a $DT$- $k$-group with a certain group operation $\ast$, where $S{C}_k^{n,l}$ is a simple closed $k$-curve with $l$ elements in ${\mathbb{Z}}^n$. Also, the $n$-dimensional digital space $({\mathbb{Z}}^n,2n,+)$ with the usual group operation " $+$" on ${\mathbb{Z}}^n$ is a $DT$- $2n$-group. Finally, the paper corrects some errors related to the earlier works in the literature.