 Solution

Idempotence gives the join order $x\le y⟺x+y=y$. Reflexivity follows from $x+x=x$; antisymmetry follows from commutativity; if $x+y=y$ and $y+z=z$, associativity gives $x+z=x+(y+z)=(x+y)+z=z$. Thus it is a partial order.