Monomial ordering problem I've got the following problem:

Let $\gamma$, $\delta$ $\in$ $\mathbb R_{> 0}$. The binary relation $\preceq$ on monomials in $X,Y$ is defined: $X^{m}Y^{n} \preceq X^{p}Y^{q}$ if and only if  $\gamma m + \delta n \leq \gamma p + \delta q .$ Show that this is a monomial ordering if and only if $ \frac \gamma \delta $ is irrational.

So since $\preceq$ is a partial order, do I just need to show that it is a total order only when $ \frac \gamma \delta $ is irrational?
Not really sure how to go about this, any guidance would be great. Thanks.

Every monomial order is a total order; in particular when $\gamma/\delta$ is a rational $a/b$, we have $X^b\preceq Y^a\preceq X^b$, so $\preceq$ is not a total order, so $\preceq$ is not a monomial order.
But not every total order is a monomial order. To verify that $\preceq$ is a monomial order you have to show:


*

*it's a total order (easy)

*$u\preceq v$ implies $uw\preceq vw$ (hint: just expand the definition of $\preceq$)

*$\preceq$ is a well-ordering (hint: show that $\{(m,n)\in\mathbb N^2\mid \gamma m+\delta n\leq L\}$ is finite for any $L$)

