What is the meaning of degree compatible ordering?

Question

Suppose I am working in a polynomial ring in several variables, say $k[X]=k[x_1,\dots, x_n]$. An ordering $<$ on $k[X]$ is said to be degree compatible, if: $\deg(X^u)<\deg(X^v) \implies X^u < X^v$ My question is that, with this definition, can lexicographic ordering be degree compatible ordering?

Answer 1

Consider $x$ and $y^2$, in the pure lexicographic order with $x>y$, $x>y^2$ while deg($x$)=1<2=deg($y^2$). If you instead consider the pure lexicographic order with $x<y$, then you can use $y$ and $x^2$ to show it is not degree compatible of your definition in your question. In general no pure lexicographic order on a polynomial with more than one variable is degree compatible of your definition. You can easily make examples similar to what we did for the case of two variables.