Polycubes and polycube ideals

To a polycube $\mathcal P$ in $\mathbb R^n$, we associate an ideal $I_{\mathcal P}$ generated by the join--meet relations of vertices that are corners of a cuboid of $\mathcal P$. For $n=2$ these are the polyomino ideals. In contrast with the planar case, we show that for $n\geq 3$ the ideal $I_{\ma