A "lattice cone" $C$ is a cone such that $x\wedge y$ and $ x\vee y$ exist for $x,y\in C$.
But how can a positive cone contain both the supremum and the infimum? Since the relation
$$x\wedge y=x+y- x\vee y$$ would imply that $x\wedge y\in C-C\neq C$.
Maybe I'm confused with the definitions, would use some help here.
0 is the minimum element of every positive cone.
A positive cone of more than one element cannot have a supremum.