Examples of order relations

Question

I am looking for some examples of order relations which are different from the quite usual ones like divisibility on $\mathbb{N}$, inclusion on a power set, "less than or equal to" ($\leq$) on $\mathbb{R}$ or lexicographic order on $\mathbb{N}$. Any suggestions? I am not interested in proving that the examples really lead to order relations (I want to try myself).

Answer 1

Let $\phi_n(x)$ be the $n$th cyclotomic polynomial (this is the monic polynomial whose zeros are the primitive $n$th roots of unity; $n$ is to be a positive integer). There is an ordering on the positive integers given by $m$ precedes $n$ if $\phi_m(x)<\phi_n(x)$ for all $x>2$.