More on the extension of linear operators on Riesz spaces

The classical Kantorovich theorem asserts the existence and uniqueness of a linear extension positive additive mapping, defined on cone $E^+$ Riesz space $E$ taking values in an Archimedean $F$, to entire $E$. We prove that, if has principal projection property $F$ is Dedekind $\sigma$-complete then for every $e \in E^+$ finitely $F$-valued measure Boolean algebra $\mathfrak{F}_e$ fragments $e$ unique ideal $E_e$ generated by $e$. If, moreover, $\tau$-continuous order continuous.