Order theory in strip folding

Despite of considerable algorithmic research on strip folding, an abstract mathematical approach to this topic is yet to be established. This paper demonstrates the possibility of constructing an ordering system within the framework of strip folding, and subsequently develops a unique systematized order theory in this context. In this study, we first utilize basic terms from order theory to arrange the relationship between folding objects and folding processes, avoiding the use of complicated terms from category theory. Next, we discuss various topics related to the representation of different folding models within this system, the meet and join operations of the preorder defined in this paper, upper sets, and their corresponding functors. In doing so, we limit our use of category theory to the minimum necessary and use a more accessible approach that utilizes fundamental concepts from order theory. Furthermore, graph theory is employed to represent the ordering system, and a sheaf structure is defined based on the graph. This methodology enables the efficient representation of strip folding in an abstract manner, thus, facilitating the exploration of various algebraic and abstract geometric structures, including sheaves and sites, among others.