Consider $A=\mathbb{Z}\times \{1\}$ with the regular order of $\mathbb{Z}$ and $B=\mathbb{Q}\times\{2\}$ with the regular order of $\mathbb{Q}$.
Prove/disprove that $A+B$ is isomorphic to $B+A$.
I think that this theorem isn't true, but after some attempts, I have failed to prove it formally.
definition: Consider $(X,\leq_{X})$ and $(Y,\leq_{Y})$ with poset. $Y$ is isomorphic to $X$ if $f:X\to Y$ is isomorphism.