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Monomial orderings (sometimes also called admissible orderings and term orderings) play a crucial on the topic of Gröbner bases, and so there is a huge literature on them. They are essentially linear orders (on the set of monomials) compatible with the multiplication.

My question is whether anybody knows some software (a function implemented for Sage, a computer program on its own, ...) that can be used to chek whether certain finite number of constraints can be satisfied in some monomial ordering.

Let me illustrate it with some examples. Suppose we consider monomials using only the variables $x,y,z,v$. Is there some software to automatically answer questions like the following ones?

  1. Is there some monomial ordering $<$ such that: $xyz < x^2 y < zv < x$ ?,

  2. Is there some monomial ordering $<$ such that: $y < xyz < zv < x y^2$ ?,

  3. etc.

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