The projective geometry over partially ordered skew fields

Derivative lattices associated with partially ordered linear spaces over partially ordered skew fields are considered. Properties of the convex projective geometry $\mathcal L$ for a partially ordered linear space ${}_FV$ over a partially ordered skew field $F$ are investigated. The convexity of linear subspaces for the linear space ${}_FV$ means the Abelian convexity ($\mathrm{ab}$-convexity), which is based on the definition of a convex subgroup for a partially ordered group. It is shown that $\mathrm{ab}$-convex directed linear subspaces plays for the theory of partially ordered linear spaces the same role as convex directed subgroups for the theory of partially ordered groups. We obtain the element-wise description of the smallest $\mathrm{ab}$-convex directed linear subspace that contains a given positive element, for a linear space over a directed skew field. It is proved that if ${}_FV$ is an interpolation linear space over a partially ordered skew field $F$, then, in the lattice $\mathcal L$, the union operation is completely distributive with respect to intersection. Properties of the projective geometry for pseudo lattice-ordered linear spaces over partially ordered skew fields are investigated.