(1+x_j) \in J_C \Leftrightarrow U_{\sigma_1} \cap (\cap_{i \in \tau_1} U^c_i) = U_{\sigma_2} \cap (\cap_{j \in \tau_2} U^c_j)$

-   $x_{i_1}+...+x_{i_m} \in J_C \Leftrightarrow$ [**for any $\sigma \subseteq [m]$ such that $|\sigma|$ is odd**]{}, $\cap_{k \in \sigma} U_{i_k} \subseteq \cup_{j \in [m] \setminus \sigma} U_{i_j}$

-   $x_{i_1} + ... + x_{i_m} + 1 \in J_C  \Leftrightarrow \cup^m_{k=1} U_{i_k} = X$ [**whenever $U_{i_1},...,U_{i_m}$ are pairwise disjoint**]{}.

By Proposition 2.6, we know that the forward direction of each of the three implications is true, so it only remains to prove the reverse implications.

[*Modified Type 4:*]{} Given a neural code $C$ that satisfies a relation of the form $U_{\sigma_1} \cap (\cap_{i \in \tau_1} U_i^c) = U_{\sigma_2} \cap (\cap_{j \in \tau_2} U_j^c) $, we will construct the desired sum by adding pseudo-monomials known to be in $J_C$. There are two cases: one where $\tau_1 \cup \tau_2$ is empty, and the remaining case where $\tau_1 \cup \tau_2$ is nonempty. We deal with each of these cases in turn.

[*Case 1 ($\tau_1 \cup \tau_2 = \emptyset$):*]{} In this case, $U_{\sigma_1} = U_{\sigma_2}$ (we ignore the trivial cases where $\sigma_1$ or $\sigma_2$ is empty or where $\sigma_1 = \sigma_2$). Then any codeword $c$ for which $\sigma_1 \subseteq c$ 