## The grading of a monomial order

Let be a field, a positive natural number, and let be an m-order on with induced monomial order on . Define the relation on by if and only if either or and .

- Prove that is an m-order. (See this previous exercise.)
- A grading of a lexicographic monomial ordering is called a
*grlex* monomial order. Sort the monomials , , , , and with respect to the lexicographic ordering and the grevlex ordering (both with ).

To show that is an m-order, we need to show that it is reflexive, antisymmetric, transitive, and total, that every nonempty subset of has a minimal element, and that respects addition.

- (Reflexive) For all , certainly . Thus .
- (Antisymmetric) Suppose and . Note that it must be the case that , and so also . Thus .
- (Transitive) Suppoe and . If the sums , , and are not all equal, then . If all three sums are equal, then in fact and , so that .
- (Total) Let . If , then ether or . If , then (since is total) either or .
- (Well-founded) Let be a nonempty subset. To each element we associate a natural number ; let be the subset of consisting of precisely those elements whose sums are minimal. Now choose an element of which is minimal with respect to . Certainly is minimal (in ) with respect to , and thus minimal in .
- (Respects addition) Suppose . If , then and we have . If , then we have , so that , and , and thus .

So is an m-order. A monomial ordering is called a *grading* if it is induced by for some m-order .

For example, note the following. Let denote the lexicographic monomial order induced by and let be the grading of .

- since
- since and lexicographically
- since
- since

Thus . Similarly, we have the following.

- since lexicographically
- since lexicographically
- since lexicographically
- since lexicographically

Thus .

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