As was mentioned above (Section 13.4.1), one needs to declare a monomial order before making a Gröbner Basis. There are various monomial orders which can be used when computing Gröbner Basis. The most common are called lexicographic and graded lexicographic orders. In the previous section, we used only graded lexicographic orders. See Section 21.1 for a discussion of lexicographic orders.
We will be considering lexicographic, graded lexicographic and multi-graded lexicographic orders. Lexicographic and multi-graded lexicographic orders are examples of elimination orderings. An elimination ordering is an ordering which is used for solving for some of the variables in terms of others.
We now discuss each of these types of orders.