Near-field structures on a given scalar group

Abstract

With this paper, we gain a better understanding of the set of near-field structures on a fixed scalar group. If we were able to describe all near-field structures on a fixed scalar group, we could describe all near-vector spaces. The near-field structures induced by isomorphisms of canonical near-vector spaces differ by quasi-multiplicative bijections while those induced by isomorphisms of near-fields differ by multiplicative bijections. This reveals one of the fundamental differences between linear algebra and near-linear algebra. We find an explicit description of all the elementary near-vector spaces. Significantly, we construct an addition $\boxplus$ on $\mathbb{Q}$ such that $(\mathbb{Q},\boxplus, \cdot)$ is isomorphic to $(\mathbb{Q}(\sqrt{-19}),+, \cdot)$. We also describe explicitly sufficient conditions for such an isomorphism to exist for more general extensions of $\mathbb{Q}$. Moreover, under extra conditions, we still describe those structures on $(\mathbb{R}, \cdot)$, and $(\mathbb{C}, \cdot)$.