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 $ (SC_k^{n, l}, \ast) $ is a $ DT $-$ k $-group with a certain group operation $ \ast $, where $ SC_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.