Cellular Binomial Ideals

- Make an ideal with $2^{n}$ cellular components, one per subset of the set of variables of $k[x_{1},\dots, x_{n}]$.
- Prove or disprove:
- Let $I$ be an $\mathcal{E}$-cellular binomial ideal. Then $I$ contains $\langle x_{i} : i \notin \mathcal{E}\rangle^{k}$ for some large enough~$k$.
- Every $\mathcal{E}$-cellular binomial ideal is of the form $J + M$, where $J \subset k[x_{i}: i \in \mathcal{E}]$ is a binomial ideal and $M \subset k[x_{i}: i\notin \mathcal{E}]$ is a monomial ideal.

- Where does the name cellular binomial ideal come from? (Hint: look at the variety).