Later, Garcia et al. discovered three more such relations, known as the Type 4-6 relations, but proved only one direction of the relations [@Grobner_bases]. It is therefore natural to ask whether the converses of any of the Type 4-6 relations might also hold.

In the sections that follow, we will prove that the answer is no for all three relations, but that we can modify the Type 4-6 relations to be if-and-only-if statements. Like the original Type 1-3 relations, the modified versions of the Type 4-6 relations are if-and-only-if at the level of both $J_C$ and a larger ideal $I(C)$, called the ideal of $C$. This suggests that, at least for the purposes of receptive field relations, the ideal and neural ideal of a code are interchangeable.

The rest of this paper will be organized as follows. In Section 2, we will introduce the background and definitions needed in the rest of the work. In Section 3, we will show by counterexample that the converses of the Type 4-6 relations do not hold. In Section 4, we will present modified versions of these relations and prove that these modified relations are if-and-only-if. Finally, in Section 5, we will discuss the 