On Weak Hypervector Spaces Over a Hyperfield

Abstract

Hypervector spaces over a hyperfield are the generalized vector spaces in which at least one operation is taken as a hyperoperation. In this paper, we provide intriguing examples and counter-examples that highlight the importance of weak hypervector spaces. Moreover, we establish the properties of subhyperspaces, direct sums, and isomorphism theorems between hypervector spaces. Also, we explore the structural aspects of weak hypervector spaces, in particular, in the case of dual spaces. Furthermore, we prove the dimension condition for annihilators in hypervector spaces.

Keywords: Hypervector spaces, Subhyperspaces, Linear transformations, Direct sum