\in \tau} U_i$ (where $\tau \neq \emptyset$), and thus $X = \cup_{i \in \tau} U_i$.

In addition, Garcia et al. found three more receptive field relationships [@Grobner_bases Theorem 5.1]:

Let $\mathcal{U} = \{U_i\}^n_{i=1}$ be a collection of sets in a stimulus space $X$. Let $C = C(\mathcal{U})$ denote the corresponding receptive field code, and let $J_C$ denote the neural ideal. Then for any subsets $\sigma_1$, $\sigma_2$, $\tau_1$, $\tau_2 \subseteq [n]$, and $m$ indices $1 \leq i_i < i_2 < ... < i_m \leq n$, with $m \geq 2$, we have receptive field relationships as follows:

-   $x_{\sigma_1} \prod_{i \in \tau_1} (1+x_i) + x_{\sigma_2} \prod_{j \in \tau_2} (1+x_j) \in J_C \Rightarrow U_{\sigma_1} \cap (\cap_{i \in \tau_1} U^c_i) = U_{\sigma_2} \cap (\cap_{j \in \tau_2} U^c_j)$.

-   $x_{i_1} + ... + x_{i_m} \in J_C \Rightarrow U_{i_k} \subseteq \cup_{j \in [m] \setminus  \{k\}} U_{i_j}$ for all $k = 1,...,m$, and if, additionally, $m$ is odd, then $\cap^m_{k=1} U_{i_k} = \emptyset$.

-   $x_{i_1} + ... + x_{i_m} + 1 \in J_C \Rightarrow \cup^m_{k=1} U_{i_k} = X$.

Notice that, unlike the Type 1-3 relations, the Type 4-6 relations are not stated as if-and-only-if statements. That is, knowing the relations among receptive fields in a stimulus space, 