The half space property for cmc $1\mathrm{/}2$ graphs in $\mathbb{E}(-1,\tau )$

In this paper, we prove a half-space theorem with respect to constant mean curvature $1\mathrm{/}2$ entire graphs in $\mathbb{E}(-\mathbb{1},\mathbb{\tau })$. If $\mathrm{\Sigma }$ is such an entire graph and ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is a properly immersed constant mean curvature $1\mathrm{/}2$ surface included in the mean convex side of $\mathrm{\Sigma }$ then ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is a vertical translate of $\mathrm{\Sigma }$. We also have an equivalent statement for the non mean convex side of $\mathrm{\Sigma }$.