Prove that there are distinct lex orders on . Show similarly that there are distinct grlex and grevlex orders.
Each lex (and grlex and grevlex) order on is (by definition) induced by a linear order of the indeterminates . Each linear order of certainly corresponds to a permutation, so that there are at most distinct lex (and grlex and grevlex) orders. It remains to be shown that distinct permutations induce distinct monomial orders.
Suppose and are distinct linear orders of . That is, there exist such that and . We immediately see that the lex, grlex, and grevlex orders induced by and are distinct, as those orders coincide with and on the set .