I was looking some latticeordered group structure. I have kind of difficulty to figure out about the group $Z^{2}$ with positive cone is $N_{>0} \times N_{>0} \cup \{(0,0)\}$ is lattice ordered group or not. Where $N_{>0}$ means all positive integer excluding zero. Thanks
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This is not a latticeordered group. As mentioned by boumol, it is (partially)ordered. A simple characterization of $(G,G_+)$ being latticeordered (where $G$ is an ordered group with positive cone $G_+$) is the following: every intersection of two translates of $G_+$ is itself a translate of $G_+$, i.e. for any $x,y \in G$, there exists $z \in G$ such that $(x + G_+) \cap (y + G_+) = z + G_+.$ (This is equivalent to being latticeordered, since this says that $z$ is the supremum of $x$ and $y$.) In your case, use $x=(0,1)$ and $y=(1,0)$ to show that your ordered group is not latticeordered. 

