Leveled partially ordered sets

Abstract

This article presents our approach to modeling a problem posed by a company aiming to optimize the 3D printing of an object within the context of additive manufacturing, where each object is created layer by layer. We introduce the concept of leveled partially ordered sets to manage large sets of points that represent each layer of the object. The core of this paper describes leveled partially ordered sets, linking them to classical ideas in ordered set theory, particularly those satisfying the Jordan–Dedekind chain condition. We also revisit and generalize Birkhoff’s concept of element height. Finally, we explore the planarity of partially ordered sets, particularly leveled partially ordered sets, and prove that under certain conditions any leveled partially ordered set can be completed into a bounded lattice without the need to add new elements to the underlying set, but merely by adding links to the original ordering relation.