Convex, balanced and absorbing subsets of hypervector spaces

Abstract

In this paper, we define convex, balanced and absorbing subsets of a hypervector space $V$ over a field $K$, where $K$ is considered $mathbb{R}$ or $mathbb{C}$ and give some examples of them. We prove that every subspace of a hypervector space is a convex and balanced subset. Also, for every regular equivalence relation $rho$ on a hypervector space $V$, we show that if $A$ is a convex, balanced or an absorbing subset of $V$, then $A/rho$ is respectively a convex, balanced or an absorbing subset of a hypervector space $V/rho$.