Digital topological groups

TL;DR

This paper develops the foundational theory of digital topological groups, categorizing them based on continuity conditions, and provides classifications, examples, and analogs of classical theorems in the digital setting.

Contribution

It introduces two categories of digital topological groups, classifies the restrictive $ ext{NP}_2$-category completely, and extends key group theory concepts to the digital context.

Findings

Complete classification of $ ext{NP}_2$-digital topological groups

Many examples of $ ext{NP}_1$-digital topological groups

Digital first isomorphism theorem established

Abstract

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.