Distinct monomial orders with same induced orderings

Abstract: We prove that the lexicographic, degree lexicographic and the degree reverse lexicographic orders for monomials in $R_n=K[X_1,...X_n]$ are uniquely determined by their induced orderings, (i.e. their restrictions to $R_{n,i}=K[X_1,..., \hat{X_i}, .., X_n]$), when $n \geq 4$. We also show that for any $n \geq 4$ there are monomial orders that are not uniquely determined by their induced orderings, and provide examples of these orders for each $n$.