$C$**]{} is $$I(C) := \{f \in  \mathbb{F}_2[x_1,...,x_n] \ | \ f(c)=0 \ \forall c \in C \}.$$

By this definition, $J_C \subseteq I(C)$. This fact will be important in later sections.

The polynomials in a neural ideal give information about the relationships among the receptive fields in a stimulus space. The first three relations (known as the Type 1-3 relations) were discovered by Curto, Itskov, Veliz-Cuba, and Youngs [@Neural_Rings]. To simplify notation, we let $$x_{\sigma} = \prod_{i \in \sigma}x_i \text{ \ \ and \ \  } U_{\sigma} = \cap_{i \in \sigma} U_i$$ for any $\sigma \in [n]$. Note that if $\sigma = \emptyset$, then $x_{\sigma} = \prod_{i \in \sigma}x_i = 1$ and $U_{\sigma} = X$.

Let $X$ be a stimulus space, let $\mathcal{U} = \{U_i\}^n_{i=1}$ be a collection of sets in $X$, and consider the receptive field code $C = C(\mathcal{U})$. Then for any pair of subsets $\sigma, \tau \subseteq [n]$, we have the following receptive field relations:

-   $x_{\sigma} \in J_C \Leftrightarrow U_{\sigma} = \emptyset$ (where $\sigma \neq \emptyset$).

-   $x_{\sigma} \prod_{i \in \tau} (1+x_i) \in J_C \Leftrightarrow U_{\sigma} \subseteq \cup_{i \in \tau} U_i$ (where $\sigma, \tau \neq \emptyset$).

-   $\prod_{i \in \tau}(1+x_i) \in J_C \Leftrightarrow X \subseteq \cup_{i 