Given a finite order ideal {mathcal {O}} in the polynomial ring K[x_1,ldots , x_n] over a field K, let partial {mathcal {O}} be the border of {mathcal {O}} and {mathcal {P}}_{mathcal {O}} the Pommaret basis of the ideal generated by the terms outside {mathcal {O}}. In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among partial {mathcal {O}}-marked sets (resp. bases) and {mathcal {P}}_{mathcal {O}}-marked sets (resp. bases). We prove that a partial {mathcal {O}}-marked set B is a marked basis if and only if the {mathcal {P}}_{mathcal {O}}-marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing partial {mathcal {O}}-marked bases and {mathcal {P}}_{mathcal {O}}-marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.