Toward an intuitive understanding of the structure of near-vector spaces

TL;DR

This paper clarifies the structure of near-vector spaces, especially over division rings, providing new insights into their decomposition, subspaces, and regularity.

Contribution

It offers a clearer understanding of near-vector spaces, characterizes regularity, and describes subspaces and spans explicitly, advancing the theoretical framework.

Findings

Simplified the definition of near-vector spaces for better understanding

Provided a new characterization of regularity in near-vector spaces

Explicitly described span and subspace structures

Abstract

In this paper, we analyze the definition Andr\'e proposed for near-vector spaces to make it more transparent. We also study the class of near-vector spaces over division rings and give a characterization of regularity that gives a new insight into the decomposition of near-vector spaces into regular subspaces. We explicitly describe span and deduce a new characterization of subspaces.