Equivariant vector bundles on toric schemes over semirings

We introduce a notion of equivariant vector bundles on schemes over semirings. We do this by considering the functor of points of a locally free sheaf. We prove that every toric vector bundle on a toric scheme $X$ over an idempotent semifield equivariantly splits as a sum of toric line bundles. We then study the equivariant Picard group $\text{Pic}_G(X)$. Finally, we prove a version of Klyachko's classification theorem for toric vector bundles over an idempotent semifield.