Neural ring homomorphism preserves mandatory sets required for open convexity

It has been studied by Curto et al. (SIAM J. on App. Alg. and Geom., 1(1) : 222 $\unicode{x2013}$ 238, 2017) that a neural code that has an open convex realization does not have any local obstruction relative to the neural code. Further, a neural code $\mathcal{C}$ has no local obstructions if and only if it contains the set of mandatory codewords, $\mathcal{C}_{\min}(\Delta),$ which depends only on the simplicial complex $\Delta=\Delta(\mathcal{C})$. Thus if $\mathcal{C} \not \supseteq \mathcal{C}_{\min}(\Delta)$, then $\mathcal{C}$ cannot be open convex. However, the problem of constructing $\mathcal{C}_{\min}(\Delta)$ for any given code $\mathcal{C}$ is undecidable. There is yet another way to capture the local obstructions via the homological mandatory set, $\mathcal{M}_H(\Delta).$ The significance of $\mathcal{M}_H(\Delta)$ for a given code $\mathcal{C}$ is that $\mathcal{M}_H(\Delta) \subseteq \mathcal{C}_{\min}(\Delta)$ and so $\mathcal{C}$ will have local obstructions if $\mathcal{C}\not\supseteq\mathcal{M}_H(\Delta).$ In this paper we study the affect on the sets $\mathcal{C}_{\min}(\Delta)$ and $\mathcal{M}_H(\Delta)$ under the action of various surjective elementary code maps. Further, we study the relationship between Stanley-Reisner rings of the simplicial complexes associated with neural codes of the elementary code maps. Moreover, using this relationship, we give an alternative proof to show that $\mathcal{M}_H(\Delta)$ is preserved under the elementary code maps.