I want to know if this makes sense:
Consider the set $\{0,1\}$ with the total order relation: $x \leq y$ if $x^y = 1$. It is reflexive, since $1^1 = 1$ and $0^0 = 1$ (I'm adopting this definition). It is vacuously antissymetric and it is transitive: $1 \leq 0, 0 \leq 0$ and $ 1 \leq 1$ are all the pairings. I'm not sure if what I'm making is OK, something feels odd to me. Please verify my possible mistakes.
As observed in the comments by @Dog_69, @Eran, @MiloBrandt, @AndreasBlass and @fleablood, it's correct, although there are more simple ways to do it, for example $x \leq' y \iff x \geq y$ or $x \leq' y \iff x | y$.