论文标题
给定标量组上的近场结构
Near-field structures on a given scalar group
论文作者
论文摘要
在本文中，我们可以更好地了解固定标量组上的一组近场结构。如果我们能够描述固定标量组上的所有近场结构，我们可以描述所有近矢量空间。典型近载体空间的同构诱导的近场结构因准多物种射击而有所不同，而近场诱导的近乎同构的生物构成的近距离结构因乘积近来的繁殖生物而异。这揭示了线性代数和近线性代数之间的基本差异之一。我们找到了所有基本近矢量空间的明确描述。值得注意的是，我们在$ \ mathbb {q} $上构造了一个添加$ \ boxplus $，这样$（\ mathbb {q}，\ boxplus，\ cdot）$是等词至$（\ mathbb {q}（q}（q}（q}））我们还描述了这种同构的明确条件，以使$ \ mathbb {q} $更一般的扩展存在。此外，在额外的条件下，我们仍然在$（\ mathbb {r}，\ cdot）$和$（\ mathbb {c}，\ cdot）$上描述这些结构。
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 $(\mathbb{C}, \cdot)$.
