Uncountable many monomial orders for monomials with 2 variables

Question

Why do we have for monomials with two variables x₁, x₂ uncountable many monomial orders?

I could prove that the order defined by

$x_1^ax_2^b ≤ x_1^cx_2^d$ is a monomial order if and only if $a + b\sqrt{2}<c + d\sqrt{2}$

Does this help for the other proof?

Thank you so much for any help!

Rolandos

Answer 1

Let $a_1,\ldots ,a_n$ be positive real numbers linear independent over the rational numbers. Now map the monomials to the real numbers by $$ f(x_1^{k_1}\cdots x_n^{k_n})=\sum_{i=1}^n k_ia_i $$ Order the monomials by the rule $\mu>\nu$ iff $f(\mu)>f(\nu)$. For $n\ge 2$ there are already uncountably many such monomials orders.