Singular

Maybe someone knows why TOP seems to have been chosen as the only module ordering to use in free resolutions of ideals and modules. I would have though the default that makes most sense as a mathematical default would have been POT, but I would rather use a signature-based ordering in the ideal case, which means each new map in the free resolution has a slightly different ordering based on the names (signatures) of the module generators. (As an aside, I am not a fan of minimizing here, just as I am not a fan of minimizing as done in normal.lib, since it is not really mathematically driven, only minimization-driven.)

You can have POT or TOP for free resolutions - depending on the ordering of the ring
(therefore a ring in singular has always also a module ordering, even if only ideals are defined in it).
There are also different algorithm to compute free resolution, some of them use internally a diffirent ordering, for example fres (Schreyer ordering or 'signature based'), lres (dp,c or TOP), etc.

Already ring r=0,(x,y,z), (c,dp); or ring r=0,(x,y,z),(C,dp); are big improvements for me. So maybe you can tell me where to look to get "signature-based" or Schreyer orderings in Singular, as I can't seem to find them mentioned in the online manual.