A weighted module view of integral closures of affine domains of type I

A type I presentation $S=R/J$ of an affine (order) domain has a weight function injective on the monomials in the footprint $\Delta(J)$. This can be extended naturally to a presentation, $\overline{R}/\overline{J}$, of the integral closure $ic(S)$. This presentation over $P$:$=F[x_n,\ldots,x_1]$ as an affine $P$-algebra relative to a corresponding grevlex-over-weight monomial ordering is shown to have a minimal, reduced Gröbner basis (for the ideal of relations $\overline{J}$) consisiting only of $P$-quadratic relations defining the multiplication of the $P$-module generators and possibly some $P$-linear relations if those generators are not independent over $P$. There then may be better choices for $P$ to minimize the number of $P$-module generators needed. The intended coding theory application is to the description of one-point AG codes, not only from curves (with $P=F[x_1]$) but also from higher-dimensional varieties.