On some partial orders associated to generic initial ideals

We study two partial orders on $[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$, the free abelian monoid on $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n$. These partial orders, which we call the strongly stable'' and the stable’’ partial order, are defined by the property that their filters are precisely the strongly stable and the stable monoid ideals. These ideals arise in the study of generic initial ideals.