Derived neighborhoods and frontier orders

We study some structural and topological properties of the frontiers of objects in a certain class of discrete spaces, in the framework of simplicial complexes and partial orders. In a previous work, we introduced the notion of frontier order, which allows to define the frontier of any object in an n-dimensional space. The main goal of this paper is to exhibit the links which exist between frontier orders and the notion of derived neighborhood as introduced in the framework of piecewise linear topology. In particular, we prove that the derived subdivision of the frontier order of an object X in a “regular” n-dimensional space is equal to the frontier of the derived neighborhood of X, and that this frontier is a union of (n− 1)-dimensional surfaces, for any dimension n.