Neural ring homomorphism preserves mandatory sets required for open convexity

Abstract

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.