Remarks on the digital-topological $k$-group structures and the development of the $A{P}_1$-$k$- and $A{P}_1^{\ast }$-$k$-group

Abstract

In the literature of a digital-topological ($DT$-, for brevity) group structure on a digital image $(X,k)$, roughly saying, two kinds of methods are shown. Given a digital image $(X,k)$, the first one, named by a $DT$-$k$-group, was established in 2022 \cite{H10} by using both the $G_{k^{\ast }}$- or $C_{k^{\ast }}$-adjacency \cite{H10} for the product $X^2:=X\times X$ and the $(G_{k^{\ast }},k)$- or $(C_{k^{\ast }},k)$-continuity for the multiplication $\alpha :X^2\to X$ \cite{H10}. The second one with the name of $N{P}_i$-$DT$-groups, $i\in \{1,2\}$, was discussed in 2023 \cite{LS1} by using the $N{P}_i(k,k)$-adjacency for $X^2$ in \cite{B1} and the $(N{P}_i(k,k),k)$-continuities of the multiplication $\alpha :X^2\to X$, $i\in \{1,2\}$. However, due to some defects of the $N{P}_u(k_1,k_2,\cdots ,k_v)$-adjacency in \cite{B1,B2}, the $A{P}_u(k_1,k_2,\cdots ,k_v)$-adjacency was recently developed as an alternative to the $N{P}_u(k_1,k_2,\cdots ,k_v)$-adjacency (see Section 4). Besides, we also develop an $A{P}_u^{\ast }(k_1,k_2,\cdots ,k_v)$-adjacency. For a digital image $(X,k)$, in case an $A{P}_1(k,k)$-($A{P}_1$-, for simplicity) adjacency on $X^2$ exists, we formulate both an $A{P}_1$-$k$- and an $A{P}_1^{\ast }$-$k$-group. Then we show that an $A{P}_1^{\ast }$-$k$-group is equivalent to a Han's $DT$-$k$-group based on both the $C_{k^{\ast }}$-adjacency on the product $X^2$ and the $(C_{k^{\ast }},k)$-continuity for the multiplication ${\alpha }_1^{\prime }:(X^2,C_{k^{\ast }})\to (X,k)$.

This paper can play an important role in the fields of general topology, digital topology, digital geometry, and so on