The close relation between border and Pommaret marked bases

Given a finite order ideal $mathcal O$ in the polynomial ring $K[x_1,dots, x_n]$ 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 $partialmathcal O$-marked sets (resp.~bases) and $mathcal P_{mathcal O}$-marked sets (resp.~bases). We prove that a $partialmathcal O$-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 $partialmathcal O$-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. Several examples are given along all the paper.