We approach the problem of discovering interesting orders in data. In many applications, it is more important to find interesting partial orders since there is often no clear ordering between certain sets of elements. Furthermore, a partial order is more robust against partially erroneous data. We present the notion of fundamental partial orders (FPO), and argue that any partial order that satisfies this property is an interesting partial order. To mine such partial orders, we present a two-stage methodology that first finds an interesting total order, and then discovers a partial order satisfying FPO using this total order. To illustrate, we focus on {0, 1} data. This is an important problem with many applications, e.g., in paleontology, where we chronologically order fossil sites by minimizing Lazarus counts. We present the experimental results of our method on paleontological data, and show that it outperforms existing approaches. The techniques developed here are general and can be abstracted for mining partial orders in any setting.