In this paper we revisit the foundations of formal concept analysis for ordered contexts of [4]. From the theoretical point of view, the obtained lattice has proved to be a proper unifying framework for reasoning about different sequential mining tasks: from the discovery of partial orders to the clustering of input sequences. Here we will show how these results on sequences can be naturally extended to the mining of any partial order structure. We empirically validate the approach by testing it on real world data. Our experimental evaluation shows that this lattice-based method is an intuitive tool for analyzing acyclic structures.