# Uncountable many monomial orders for monomials with 2 variables

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}

Does this help for the other proof?

Thank you so much for any help!

Rolandos

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.