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.