Nested set complexes for posets and the Bier construction

We generalize the concept of combinatorial nested set complexes to posets and exhibit the topological relationship between the arising nested set complexes and the order complex of the underlying poset. In particular, a sufficient condition is given so that this relationship is actually a subdivision. We use the results to generalize the proof method of \v{C}uki'c and Delucchi, so far restricted to semilattices, for a result of Bj"orner, Paffenholz, Sj"ostrand and Ziegler on the Bier construction on posets.