On the Inheritance of Orbifold Substructures

In a previous article, we defined a very flexible notion of suborbifold and characterized those suborbifolds which can arise as the images of orbifold embeddings. In particular, suborbifolds are images of orbifold embeddings precisely when they are saturated and split. This article addresses the problem of orbifold structure inheritance for three orbifolds $\mathcal{Q}\subset\mathcal{P}\subset\mathcal{O}$. We identify an appealing but ultimately inadequate notion of an inherited canonical orbifold substructure. In particular, we give a concrete example where the orbifold structure of $\mathcal{Q}$ is canonically inherited from $\mathcal{P}$, and the orbifold structure of $\mathcal{P}$ is canonically inherited from $\mathcal{O}$, but the orbifold structure of $\mathcal{Q}$ is not canonically inherited from $\mathcal{O}$. On the other hand, it is easy to see that when $\mathcal{Q}$ is embedded in $\mathcal{P}$, and $\mathcal{P}$ is embedded in $\mathcal{O}$, all of the canonical inherited orbifold substructures will agree. We also investigate the property of saturation in this context, and give an example of a suborbifold with the canonical orbifold substructure that is not saturated.