Digital topological groups
This work addresses a theoretical gap in digital topology for mathematicians, but it is incremental as it extends existing concepts to a new setting without broad practical applications.
The paper tackles the problem of developing a foundational theory for digital topological groups by defining two categories (NP1 and NP2) based on continuity requirements, resulting in a complete classification of NP2 groups and examples for NP1 groups, along with digital versions of homomorphisms and the first isomorphism theorem.
In this article, we develop the basic theory of digital topological groups. The basic definitions directly lead to two separate categories, based on the details of the continuity required of the group multiplication. We define $\NP_1$- and $\NP_2$-digital topological groups, and investigate their properties and algebraic structure. The $\NP_2$ category is very restrictive, and we give a complete classification of $\NP_2$-digital topological groups. We also give many examples of $\NP_1$-digital topological groups. We define digital topological group homomorphisms, and describe the digital counterpart of the first isomorphism theorem.