Cellular Binomial Ideals
  1. Make an ideal with $2^{n}$ cellular components, one per subset of the set of variables of $k[x_{1},\dots, x_{n}]$.
  2. Prove or disprove:
    1. 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~$k$.
    2. 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.
  3. Where does the name cellular binomial ideal come from? (Hint: look at the variety).
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