An ordering of monomials in the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ is called a monomial ordering (EDIT: it seems that an equally common term used in this context is "term ordering", while "monomial ordering" only requires the first condition below) if

it is a well-ordering;

for three monomials $a,b,c$ we have $ac<bc$ and $ca<cb$ whenever $a<b$.

My main general question is whether there are any general classification results for monomial orderings. A very mild particular case which is of interest already would be to assume that our ordering extends the partial order given by the total degree.

I spent some time thinking about it, and realised that I lack good intuition about it. In the commutative case, the situation is quite straightforward: every order is an appropriate superposition of several partial orderings given by assigning some weights to generators, as proved by L. Robbiano in the paper "Term orderings on the polynomial ring" MR0826583 (87e:13006). In the noncommutative case, I do not expect any description which is remotely as economic (though who knows), but even partial results could be helpful.

My motivation, as it probably always the case, is coming from Gröbner bases: there are several situations where I could guess a good ordering solving some problem, but there are cases where my imagination is not enough, and I would like to know if there are ways to look for a "good" ordering in a way which is as far from mere guessing as possible.

rightorderings were discussed, and some other papers discussing orderings of groups which are very far from free. $\endgroup$