DT-k-group structures on digital objects and an answer to an open problem.

The paper examines various properties of D T - k -subgroup structures and addresses an open problem on the existence of topological group structures on the n -dimensional Khalimsky (K -, for brevity) topological space and Marcus–Wyse (M -, for short) topological plane. In particular, we obtain many types of totally k -disconnected or k -connected subgroups of a k -connected D T - k -group. Besides, we prove that each of the n -dimensional K -topological space and the M -topological plane cannot be a typical topological group. Unlike an existence of a D T - k -group structure of (S C k n , l , ∗) (see Proposition 4.7), we prove that neither of (S C K n , l , ∗) and (S C M l , ∗) is a topological group, where S C K n , l (respectively, S C M l ) is a simple closed K - (respectively, M -) topological curve with l elements in ℤ n (respectively, ℤ 2 ) and the operation "∗" is a special kind of binary operation for establishing a group structure of each of S C K n , l and S C M l . Finally, given a D T - k -group structure of (S C k 1 n 1 , l 1 × S C k 2 n 2 , l 2 , ∗) , we find several types of D T - k -subgroup structures of it (see Theorem 5.7). [ABSTRACT FROM AUTHOR]