Multiplicative near-vector spaces
L. Boonzaaier, S. Marques, D. Moore
TL;DR
This paper addresses the scarcity of explicit examples in near-vector space theory by introducing a broad, computable family of near-vector spaces built from multiplicative automorphisms of near-fields. It develops the structure $F^{\boldsymbol{\sigma},\boldsymbol{\rho}}$ of multiplicative near-vector spaces, proves that $Q(F^{\boldsymbol{\sigma},\boldsymbol{\rho}})=F$ and establishes regular decompositions and several equivalent characterizations, including the isomorphisms to canonical forms $F^{\boldsymbol{\theta},\boldsymbol{\mathrm{Id}}}$ or $F^{\boldsymbol{\mathrm{Id}},\boldsymbol{\theta}}$. It then analyzes infinite products, showing that infinite direct products preserve the near-vector space structure for finite-field-based setups under finiteness conditions, while real and complex cases can fail, and finally introduces complexification of real multiplicative near-vector spaces. Together, these results provide concrete tools for constructing and manipulating near-vector spaces and pave the way for applications in non-linear algebraic contexts.
Abstract
Near-vector spaces extend linear algebra tools to non-linear algebraic structures, enabling the study of non-linear problems. However, explicit constructions remain rare. This paper introduces a broad computable family of near-vector spaces, called multiplicative, and explores their properties. This family is fully determined over finite, real, and complex fields. We also discuss the existence of infinite coproducts, and products in the category of near-vector spaces. Finally, we introduce the complexification of a multiplicative near-vector space over the real numbers.